JKd^^I.    Q»/^. 


LIBRARY         Ex«u*«ri»« 

library 


OF  THE 


UNIVERSITY  OF  CALIFORNIA. 

Accession  No. /^hJlfS..   Class  No. 


Digitized  by  tine  Internet  Archive 

in  2007  with  funding  from 

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http://www.archive.org/details/elementsofdifferOObyerrich 


ELEMENTS 


DIFFERENTIAL  CALCULUS. 


WITH 


EXAMPLES   AND  APPLICATIONS. 


A    TEXT    BOOK 


BY 


W.   E.   BYERLY,    Ph.D., 

ASSISTANT   PROFESSOR  OF   MATHEMATICS   IN    HARVARD   UNIVERSITY. 


-*  .i.^  ^  K 


DIVERSITY 

BOSTON,    U.S.A.: 
PUBLISHED  BY   GINN  &  COMPANY, 

1898. 


15 


Kngineeriiig 
Library 


Entered  according  to  Act  of  Congress,  in  tiie  year  1879,  by 

W.  E.  BYERLY, 
in  the  oflttce  of  tiie  Librarian  of  Congress,  ai  Washington. 


Ttpographt  bt  J.  S.  CusHiNo  &  Co.,  Boston. 

t'BESSWORK    BT   GiNN   &  Co.,   BOSTON. 


PREFACE. 


The  following  book,  which  embodies  the  results  of  m}-  own 
experience  in  teaching  the  Calculus  at  Cornell  and  Harvard 
Universities,  is  intended  for  a  text -book,  and  not  for  an 
exhaustive  treatise. 

Its  peculiarities  are  the  rigorous  use  of  the  Doctrine  of 
Limits  as  a  foundation  of  the  subject,  and  as '  preliminary 
to  the  adoption  of  the  more  direct  and  practically  convenient 
infinitesimal  notation  and  nomenclature  ;  the  early  introduc- 
tion of  a  few  simple  formulas  and  methods  for  integrating; 
a  rather  elaborate  treatment  of  the  use  of  infinitesimals  in 
pure  geometry ;  and  the  attempt  to  excite  and  keep  up  the 
interest  of  the  student  by  bringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  numerous  applications  to 
practical  problems  in  geometr}^  and  mechanics. 

I  am  greatly  indebted  to  Prof.  J.   M.  Peirce,  from  whose 

lectures  I  have  derived  many  suggestions,  and  to  the  works 

of  Benjamin  Peirce,  Todhunter,  Duhamel,  and  Bertrand,  upon 

which  I  have  drawn  freely. 

W.    E.    BYERLY. 

Cambridge,  October,  1879. 


TABLE   OF   CONTENTS. 


CHAPTER  I. 


INTRODUCTION. 
Article.  Page 

1.  Be^mtion  of  variable  &nd  constant 1 

2.  Defiuition  of  function  and  independent  variable 1 

3.  Symbols  by  which  fimctional  dependence  is  expressed    ....  2 

4.  Defiuition  of  increment.     Notation  for  an  increment.     An  in- 

crement may  be  positive  or  negative 2 

5.  Definition  of  the  limit  of  a  variable 3 

6.  Examples  of  limits  in  Algebra 3 

7.  Examples  of  limits  in  Geometry 4 

8.  The  fundamental  proposition  in  the  Theory  of  Limits 5 

9.  Application  to  the  proof  of  the  theorem  that  the  area  of  a  circle 

is  one-half  the  product  of  the  circumference  by  the  radius .    .  5 

10.  Importance  of  the  clear  conception  of  a  limit 6 

11.  The  velocity  of  a  moving  body.     Mean  velocity  ;  actual  velocity 

at  any  instant ;  uniform  velocity ;  variable  velocity 6 

12.  Actual  velocity  easily  indicated  by  aid  of  the  increment  notation  7 

13.  Velocity  of  a  falling  body 7 

14.  The  direction  of  the  tangent  at  any  point  of  a  given  curve. 

De^mtion  of  tangent  as  limiting  case  of  secant 8 

15.  The  inclination  of  a  curve  to  the  axis  of  X  easily  indicated  by 

the  aid  of  the  increment  notation 8 

16.  The  inclination  of  a  parabola  to  the  axis  of  A'^ 9 

17.  Fundamental  object  of  the  Difl'erential  Calculus    .......  10 

CHAPTER  II. 

DIFFERENTIATION   OF   ALGEBRAIC   FUNCTIONS. 

18.  jye^mtion  of  derivative.     Derivative  of  a  co7is^a>i? 11 

19.  General  method  of  finding  the  derivative  of  any  given  function. 

General  formula  for  a  derivative.    Examples 11 


VI  DIFFERENTIAL  CALCULUS. 

Article.  Page. 

20.  Classification  of  functions 12 

21.  Differentiatiou  of  the  product  of  a  constant  and  the  variable  ;  of 

a  poicer  of  the  variable,  where  the  exponent  is  a  positive 

integer 13 

22.  Derivatixe  of  a  sum  of  functions 14 

23.  Deriyative  of  SL  product  of  functions 15 

24.  Deriv'dtiye  of  a  quotient  of  functions.    Examples 17 

25.  Derivative  of  a  function  of  a  function  of  the  variable 18 

26.  Derivative  of  a  power  of  the  variable  where  the  exponent  is 

negative  or  fractional.     Complete  set  of  formulas  for  the 
differentiation  of  Algebraic  functions.     Examples 19 


CHAPTER  III. 

APPLICATIONS. 

Tangents  and  Normals. 

27.  Direction  of  ia^i^feni  and  norma?  to  a  plane  curve .    22 

28.  Equations  of  tangent  and  normal.     Subtangent.     Subnormal. 

Length  of  tangent.     Length  of  normal.     Examples 23 

29.  Derivative  may  sometimes  be  found  by  solving  an  equation. 

Examples 25 

Indeterminate  Forms. 

30.  Definition  of  infinite  and  infinitely  great 26 

31.  Value  of  a  function  corresponding  to  an  infinite  value  of  the 

variable 26 

32.  Infinite  value  of  a  function  corresponding  to  a  particular  value 

of  the  variable 27 

33.  The  expressions  Q,  ^,  and  OXoo,  called  indeterminate  forms. 

When  definite  values  can  be  attached  to  them 28 

34.  Treatment  of  the  form  ^.     Examples 28 

35.  Reduction  of  the  forms  g  and  0  X  oo  to  the  form  ^ 30 

Maxima  and  Minima  of  a  Continuous  Function. 

36.  Con«uiMO«s  change.     Continuous  function 31 

37.  If  a  function  increases  with  the  increase  of  the  variable,  its 

derivative  is  positive ;  if  it  decreases,  negative 31 

38.  Value  of  derivative  shows  rate  of  increase  of  function    ....    32 

39.  Definition  of  maximum  and  minimum  values  of  a  function     .   .    32 


TABLE   OF   CONTENTS.  vii 

Article.  Page. 

40.  Derivative  zero  at  a  maximum  or  a  minimum 33 

41.  Geometrical  illustration 33 

42.  Sign  of  derivative  near  a  zero  value  shown  by  the  value  of  its 

own  derivative 34 

43.  Derivatives  of  different  orders 34 

44.  Numerical  example 34 

45.  Investigation  of  a  minimum 35 

46.  Case  where  the  third  derivative  must  be  used.     Examples     .    .  35 

47.  General  rule  for  discovering  maxima  and  minima.     Examples  .  36 

48.  Use  of  auxiliary  variables.     Examples 38 

49.  Examples 39 

Integration. 

50.  Statement  of  the  problem  of  finding  the  distance  traversed  by  a 

falling  body,  given  the  velocity 41 

51.  Statement  of  the  problem  of  finding  the  area  bounded  by  a  given 

curve 41 

52.  Statement  of  the  problem  of  finding  the  length  of  an  arc  of  a 

given  curve 42 

53.  Integration.     Integral 44 

54.  Arbitrary  constant  in  integration 44 

55.  Some  formulas  for  direct  integration 44 

56.  Solution  of  problem  stated  in  Article  50 ..45 

57.  Example  under  problem  stated  in  Article  51.     Examples    ...  46 

58.  Examples  under  problem  stated  in  Article  52 48 


CHAPTER   IV. 

TRANSCENDENTAL   FUNCTIONS. 

59.  Differentiation  of  \ogx  requires  the  investigation  of  the  limit 

0^1+3" *' 

60.  Expansion  of  (  1  +  —  )    '\v  the  Binomial  Theorem 50 

61.  Proof  that  the  limit  in  question  is  the  sum  of  a  well-known 

series : 50 

62.  This  series  is  taken  as  the  base  of  the  natural  system  of  loga- 

rithms.   Computation  of  its  numerical  value 52 

63.  Extension  of  the  proof  given  above  to  the  cases  where  m  is  not 

a  positive  integer 53 

64.  Differentiation  of  logo:  completed 54 

65.  Differentiation  of  a*.     Examples 55 


VIU  DIFFERENTIAL   CALCULUS. 

Trigonometric  Functions. 

Article.  Fag& 

66.  Circular  measure  of  an  angle.    Reduction  from  degree  to  cir- 

cular m^diSMve.     Value  of  the  inuHu  circular  measure    ...    57 

67.  Differentiation  of  siux  requires  the  investigation  of  the  limit 

sin  Aa;       ,  1  — cos  Ax 

— - —  and 57 

Aa:  Ax 

68.  Investigation  of  these  limits 58 

69.  Differentiation  of  the  Trigonometric  Functions.     Examples     .  59 

70.  Anti-  or  inverse  Trigonometric  Functions 60 

71.  Differentiation  ofthe  Anti-Trigonometric  Functions.   Examples  60 

72.  Anti-  or  inverse  notation.     Differentiation  of  anti-  functions  in 

general 61 

73.  The  derivative  of  y  with  respect  to  x,  and  the  derivative  of  x 

vv^ith  respect  to  y,  are  reciprocals.     Examples G2 


CHAPTER  V. 

INTEGRATION. 

74.  Formulas  for  direct  integration 65 

75.  Integration  by  substitution.     Examples 66 

76.  If /x  can  be  integrated,  f(a  +  bx)  can  always  be  integrated.   Ex- 

amples       67 

^^'  -^^sjia'-x'y     ^^^"^Pl^s 67 

78.  L^^J^^,y    Example 68 

79.  Integration  by  parts.     Examples 69 

80.  /xSin-'x.     Examples 69 

81.  Use  of  integration  by  substitution  and  integration  by  parts  in 

combination.     Examples 70 

82.  Simplification  by  an  algebraic  transformation.     Examples  ...  71 

Applications. 

83.  Area  of  a  segment  of  a  circle ;  of  an  ellipse ;  of  an  hyperbola   .  72 

84.  Length  of  an  arc  of  a  circle 74 

85.  Length  of  an  arc  of  a  parabola.     Example ...  75 


CHAPTER   VI. 

CURVATURE. 

86.    Total  curvature ;  mean  curvature ;  actual  curvature.     Formula 

for  actual  curvature 77 


TABLE   OF   CONTENTS.  ix 
Article.                                                                                                                                                       Page. 

87.  To  find  actual  curvature  conveniently,  an  indirect  method  of 

difl'erentiation  must  be  used 77 

88.  The  derivative  of  z  with  respect  to  y  is  the  quotient  of  the 

derivative  of  z  with  respect  to  x  by  the  derivative  of  y  with 

respect  to  x 78 

89.  Reduced  formula  for  curvature.     Examples 78 

90.  Osculating  circle.     Badius  of  curvature.     Centre  of  curvature   .  81 

91.  Definition  of  evolute.     Formulas  for  e volute 82 

92.  Evolute  of  a  parabola 83 

93.  Reduced  formulas  for  evolute.     Example 85 

94.  Evolute  of  an  ellipse.     Example 85 

95.  Every  normal  to  a  curve  is  tangent  to  the  evolute 87 

96.  Length  of  an  arc  of  evolute 88 

97.  Derivation  of  the  name  evolute.     Involute 88 


CHAPTER  VII. 

THE    CYCLOID. 

98.  Definition  of  the  cycloid 90 

99.  Equations  of  cycloid  referred  to  the  base  and  a  tangent  at  the 

lowest  point  as  axes.     Examples 90 

100.  Equations  of  the  cycloid  referred  to  vertex  as  origin.     Exam- 

ples       92 

101.  Statement  of  properties  of  cycloid  to  be  investigated    ....  93 

102.  Direction  of  tangent  and  normal.     Examples 93 

103.  Equations  of  tangent  and  normal.     Example 94 

104.  Subtangent.     Subnormal.     Tangent.     Normal 94 

105.  Curvature.     Examples      95 

106.  Evolute  of  cycloid     96 

107.  Length  of  an  arc  of  cyloid 97 

108.  Area  of  cycloid.     Examples 98 

109.  Definition  and  equations  of  epicycloid  and  hypocycloid.    Exam- 

ples       99 


CHAPTER   VIII. 

PROBLEMS   IN  MECHANICS. 

110.  Formula  for  velocity  in  terms  of  distance  and  time '.  102 

111.  Acceleration.     Example.     Differential  equations  of  motion  .    .102 

112.  Two  principles  of  mechanics  taken  for  granted 103 

113.  Problem  of  a  body  falling  freely  near  the  earth's,  surface  .   .   .103 


X  DIFFERENTIAL   CALCULUS. 

Article.  Page. 

114.  Second  method  of  integrating  the  equations  of  motion  in  the 

case  of  a  falling  body 104 

115.  Motiondowu  an  inclined  plane  .  ^ 106 

116.  Motion  of  a  body  sliding  down  a  chord  of  a  vertical  circle. 

Example 107 

117.  Problem  of  a  body  falling  from  a  distance  toward  the  earth. 

Velocity  of  fall.     Limit  of  possible  velocity.     Time  of  fall. 
Examples 108 

118.  Motion  down  a  smooth  curve.     Examples 112 

119.  Time  of  descent  of  a  particle  from  any  point  of  the  arc  of  an 

inverted  cycloid  to  the  vertex.     Cycloidal  pendulum.     Tau- 
tochrone 113 

120.  A  problem  for  practice 116 


CHAPTER  IX. 

DEVELOPMENT   IN   SERIES. 

121.  Definition  of  series.      Convergent  series.      Divergent  series. 

Sum  of  series 117 

122.  Example  of  a  series 117 

123.  Function  obtained  by  integration  from  one  of  its  derivatives. 

Series  suggested 118 

124.  Development  of  /(x„+  h)  into  a  series.     Determination  of  the 

coefficients  on  the  assumption  that  the  development  is  pos- 
sible.    Examples 119 

125.  An  error  committed  in  taking  a  given  number  of  terms  as 

equivalent  to  the  function  developed 122 

126.  Lemma 122 

127.  Error  determined 123 

128.  Second  form  for  remainder.     Taylor's  Theorem 125 

129.  Examples  of  use  of  expression  for  remainder.     Test  for  the 

possibility  of  developing  a  function 126 

130.  Development  of  log  (1  + a;) 127 

131.  The  Binomial  Theorem.     Investigation  of  the  cases  in  which 

the  ordinary  development  holds  for  a  negative  of  fractional 
value  of  the  exponent.     Example 129 

132.  Maclaurin's  Theorem 132 

133.  Development  of  a^  e^  and  e 133 

134.  Development  of  sin  x  and  cos  x 134 

135.  Development  of  sin-^  a;  and  tan-^  X.     Examples 134 

136.  The  investigation  of  the  remainder  in  Taylor's  Theorem  often 

omitted.    Examples 136 


TABLE   OF   CONTENTS.  Xi 

Article.  Page. 

137.  Leibnitz's  Theorem  for  Derivatives  of  a  Product 136 

138.  Development  of  tan  x.    Example 137 

Indeterminate  Forms. 

139.  Treatment  of  indeterminate  forms  by  the  aid  of  Taylor's  The- 

orem.    The  form  q.     Example 138 

140.  Special  consideration  of  the  case  where  the  form  ^  occurs  for 

an  infinite  value  of  the  variable 139 

141.  The  form  ^ ;  special  consideration  of  the  cases  where  its  true 

vahie  is  zero  or  infinite 141 

142.  Reduction  of  the  forms  oo^,  1*,  0^  to  forms  already  discussed. 

Examples      142 

Maxima  and  Minima. 

143.  Treatment  of  mao^ma  rt?if?  ??ii»ma  by  Taylor's  Theorem  .   .    .  145 
145.   Generalization  of  the  investigation  in  the  preceding  article. 

Examples 146 


CHAPTER  X. 

INFINITESIMALS. 

146.  Definition  of  infinitesimal 149 

147.  Principal  infinitesimal.     Order  of  an  infinitesimal.    Examples  .  149 

148.  Determination  of  the  order  of  an  infinitesimal.     Examples  .    .  150 

149.  Infinitesimal  increments  of  a  function  and  of  the  variable  on 

which  it  depends  are  of  the  same  order 151 

150.  Lemma.     Expression  for  the  the  coordinates  of  points  of  a 

curve  by  the  aid  of  an  auxiliary  variable 152 

151.  Lemma 153 

152.  Lemma 153 

153.  Geometrical  example  of  an  infinitesimal  of  the  second  order    .  154 

154.  In  determining  a  tangent  the  secant  line  can  be  replaced  by  a 

line  infinitely  near 155 

155.  Tangent  at  any  point  of  the  pec?a/ of  a  given  curve 156 

156.  The  locus  of  the  foot  of  a  perpendicular  let  fall  from  the  focus 

of  an  ellipse  upon  a  tangent.     Example 157 

157.  Tangent  at  any  point  of  the  locus  of  a  point  cutting  off  a  given 

distance  on  the  normal  to  a  given  curve 158 

168.    Tangent  to  the  locus  of  the  vertex  of  an  angle  of  constant 

magnitude,  circumscribed  about  a  given  curve.     Example  .  159 


XU  DIFFERENTIAL   CALCULUS. 

Artiole.  Page. 

159.  The  substitution  of  one  infinitesimal  for  another 160 

160.  Theorem  concerning  the  limit  of  the  ratio  of  two  infinitesi- 

mals    160 

161.  Theorem  concerning  the  limit  of  the  sum  of  infinitesimals  .    .  161 

162.  If  two  infinitesimals  difler  from  each  other  by  an  infinitesimal 

of  higher  order,  the  limit  of  their  ratio  is  unity 162 

163.  Direction  of  a  tangent  to  a  parabola 163 

164.  Area  of  a  sector  of  a  parabola 164 

165.  The  limit  of  the  ratio  of  an  infinitesimal  arc  to  its  chord  is 

unity 165 

166.  Rough  use  of  infinitesimals 166 

167.  Tangent  to  an  ellipse.     Examples 167 

168.  The  area  of  a  segment  of  a  parabola.     Examples 168 

169.  New  way  of  regarding  the  cycloid 169 

170.  Tangent  to  the  cycloid 170 

171.  ^rea  of  the  cycloid 171 

172.  Length  of  an  arc  of  the  cycloid 172 

173.  Hadius  of  curvature  of  the  cycloid 174 

174.  ^voZif^e  of  the  cycloid 176 

175.  Examples 177 

176.  The  brachistochrone,  or  curve  of  quickest  descent  a  cycloid     .   .  177 


—      CHAPTER   XI. 

DIFFERENTIALS. 

177.  In  obtaining  a  derivative  the  increment  of  the  function  may  be 

replaced  by  a  simpler  infinitesimal.     Application  to  the  de- 
rivative of  an  area  ;  to  the  derivative  of  an  arc 183 

178.  Definition  of  differential 185 

179.  Difierential  notation   for  a  derivative.     A  derivative   is   the 

actual  ratio  of  two  diff*erentials 185 

180.  Advantage  of  the  difierential  notation 185 

181.  Formulas  for  difiierentials  of  functions.     Examples 186 

182.  The  differential  notation  especially  convenient  in  dealing  with 

problems  in  integration.    Numerical  example 187 

183.  Integral  regarded  as  the  limit  of  a  sum  of  differentials.     Defi- 

nite integral 188 

184.  An  area  regarded  as  the  limit  of  a  sum  of  infinitesimal  rectan- 

gles.    Example 189 

186.   Definition  of  centre  of  gravity.     The  centre  of  gravity  of  a 

parabola 190 


TABLE   OF   CONTENTS.  XUl 

Differentials  of  Different  Orders. 

Article.  Page. 

186.  Definition  of  the  order  of  a  differential 192 

187.  Relations  between  differentials  and  derivatives  of  different 

orders.    Assumption  tliat  the  differential  of  the  independent 
variable  is  constant 193 

188.  Expression  for  the  second  derivative  in  terms  of  differentials 

when  no  assumption  is  made  concerning  the  differential  of 
the  independent  variable 194 

189.  Differential  expression  for  the  radius  of  curvature 194 

190.  Finite  differences  or  increments  of  different  orders 194 

191.  Any  infinitesimal  increment  differs  from  the  differential  of  the 

same  order  by  an  infinitesimal  of  higher  order 196 

192.  Lemma 197 

193.  Proof  of  statement  in  Article  191 197 


CHAPTER  XII. 

FUNCTIONS   OF  MORE  THAN   ONE   VARIABLE. 

Partial  Derivatives. 

194.  l\\\x^tY2it\ou  o^  Si  function  of  two  variables 199 

195.  Definition  of  partial  derivative  of  a  function  of  several  varia- 

bles.    Illustration 199 

196.  Successive  partial  derivatives 200 

197.  In  obtaining  successive  partial  derivatives  the  order  in  which 

the  differentiations  occur  is  of  no  consequence 200 

198.  Complete  differential  of  a  function  of  two  variables.    Example  .  202 

199.  Use  of  partial  derivatives  in  obtaining  ordinary  or  complete 

derivatives.     Example 203 

200.  Special  case  of  Article  199.     Examples 204 

201.  Use  of  partial  derivatives  in  finding  successive  complete  deriv- 

atives.    Example 205 

202.  Derivative  of  an  implicit  function.     Examples 206 

203.  Second  derivative  of  an  implicit  function.    Examples  ....  207 


CHAPTER  XIII. 

CHANGE   OF   VARIABLE. 

204.  If  the  independent  variable  is  changed,  differentials  of  higher 

orders  than  the  first  must  be  replaced  by  more  general 
values . 209 

205.  Example , 209 


XIV  DIFFERENTIAL   CALCULUS. 

Article.  Page. 

206.  Example  of  the  change  of  both  dependent  and  independent 

variable  at  the  same  time.     Example 211 

207.  Direction  of  a  tangent  to  a  curve  in  terms  of  polar  coordi- 

nates.    Examples 213 

208.  Treatment  of  the  subject  of  change  of  variable  without  the  use 

of  differentials.     Example 214 

209.  Change  of  variable  when  partial  derivatives  are  employed. 

First  Method.     Examples 215 

210.  Second  Method.     Examples 217 

211.  Third  Method.     Examples 218 


CHAPTER  XIV. 

TANGENT  LINES   AND    PLANES  IN   SPACE. 

212.  A  curve  in  space  is  represented  by  a  pair  of  simultaneous 

equations 220 

213.  Equations  of  tangent  line  to  a  curve  in  space.     Equation  of 

normal  plane 220 

214.  Tangent  and  normal  to  helix 221 

215.  Expressions  for  equation  of  tangent  line  to  curve  in  space  in 

terms  of  partial  derivatives.     Examples 224 

216.  Osculating  plane  to  3i  cuYwc  in  s^nce.     Example 225 

217.  Tangent  plane  to  Si  surface.     Examples .226 


CHAPTER  XV. 

DEVELOPMENT    OF   A   FUNCTION   OP    SEVERAL    VARIABLES. 

218.  Taylor's  and  3Iaclaurin's  Theorems  for  functions  of  two  inde- 

pendent variables.     Example 227 

219.  Taylor's  Theorem  for  three  variables.     Example 231 

220.  Euler's  Theorem  fov  homogeneous  functions.     Example     .    .    .232 

CHAPTER  XVI. 

MAXIMA  AND  MINIMA  OF  FUNCTIONS  OF  TWO  OR  MORE  VARIABLES. 

221.  Essential   conditions   for  the   existence    of  a  maximum  or 

minimum 234 

222.  Tests  for  the  detection  of  maxima  and  minima 235 

223.  Formulas  for  maxima  and  minima 236 

224.  Examples 236 

225.  Examples 239 


TABLE  OF  CONTENTS.  XV 

CHAPTER  XVII. 

THEORY   OF   PLANE  CURVES. 

Concavity  and  Convexity. 

Article.  Page. 

226.  Tests  for  concau%  and  convex%  of  plane  curves 240 

227.  Points  of  inflection 240 

228.  Application  of  Taylor's  Theorem  to  the  treatment  of  convexity 
and  concavity  and  points  of  inflectiou 241 

229.  Examples 243 

230.  Singular  points 245 

231.  Be&nitlon  of  multiple  point ;  osculating  point ;  cusp;  conjugate 

point ;  point  d' arret ;  point  saillant 245 

232.  Test  for  a  multiple  point 246 

233.  Detection  of  oscidating  points,  cusps,  conjugate  points,  points 

d'arret,  and  points  saillant 247 

234.  Example  of  a  double  point 247 

235.  Example  of  a  C!<sp 249 

236.  Example  oT  SL  conjugate  point.     Examples 250 

Contact  of  Curves. 

237.  Order's  of  contact 251 

238.  Order  of  contact  indicates  closeness  of  contact 252 

239.  Oseulating  circle.     Examples 253 

Envelops. 

240.  An  equation    may   represent   a  series  of   curves.      Variable 

parameter.     Envelop 255 

241.  Determination  of  the  equation  of  an  enveZop 255 

242.  Example 257 

243.  An  evolute  the  envelop  of  the  normal.     Examples 257 


DIFFEKENTIAL  CALCULUS. 


CHAPTER   I. 

INTRODUCTION. 


1.  A  variable  quantity^  or  simply  a  variable^  is  a  quantity 
wliich,  under  tlie  conditions  of  the  problem  into  which  it  enters, 
is  susceptible  of  an  indefinite  number  of  values. 

A  constant  quantity^  or  simply  a  constarit,  is  a  quantity  which 
has  a  fixed  value. 

For  example  ;  in  the  equation  of  a  circle 

«2  _(_  2/2  =  a\ 

X  and  y  are  variables,  as  thej'  stand  for  the  coordinates  of  any 
point  of  the  circle,  and  so  may  have  any  values  consistent  with 
that  fact ;  that  is,  they  may  have  an  unlimited  number  of  different 
values ;  a  is  a  ^constant,  since  it  represents  the  radius  of  the 
circle,  and  has  therefore  a  fixed  value.  Of  course,  any  given 
number  is  a  constant. 

2.  When  one  quantity  depends  upon  another  for  its  value,  so 
that  a  change  in  the  second  produces  a  change  in  the  first,  the 
first  is  called  a  function  of  the  second.  If,  as  is  generall}"  the 
case,  the  two  quantities  in  question  are  so  related  that  a  change 
in  either  produces  a  change  in  the  other,  either  may  be  regarded 
as  a  function  of  the  other.  The  one  of  which  the  other  is 
considered  a  function  is  called  the  independent  variable^  or 
simply  the  vario^ble. 


2  DIFFERENTIAL  CALCULUS.  [Art.  3. 

For  example ;  if  x  and  y  are  two  variables  connected  by  the 
relation 

y  =  x^, 

we  may  regard  x  as  the  independent  variable,  and  then  y  will 
be  a  function  of  x,  for  an}'  change  in  x  produces  a  corresponding 
change  in  its  square  ;  or  we  may  regard  y  as  the  independent 
variable,  and  then  x  will  be  a  function  of  y,  and  from  that  point 
of  view  the  relation  would  be  more  naturally  written 

x=  'sfy. 

Again,  suppose  the  relation  is 

y  =  sin  ic, 

we  may  either  regard  ?/  as  a  function  of  x,  in  which  case  we 
should  naturally  write  the  relation  as  above,  or  we  may  regard  x 
as  a  function  of  y,  and  then  we  should  more  naturally  express 
the  same  relation  by 

X  =  sin~^2/, 

i.e.,  X  is  equal  to  the  angle  whose  sine  is  y. 

3.  Functional  dependence  is  usually  indicated  by  the  letters 
/,  jF,  and  (f.  Thus  we  may  indicate  that  y  is  a  function  of  x 
by  writing 

y  =  fx,  or  y  =  Fx,  or  y=<fx; 

and  in  each  of  these  expressions  the  letter  /,  F,  or  <p  is  not  an 
algebraic  quantit}^,  but  a  mere  sj^mbol  or  abbreviation  for  the 
word  function^  and  the  equation  is  precisely  equivalent  to  the 
sentence,  y  depends  upon  x  for  its  value ^  so  that  a  chayige  in  the 
value  of  X  will  necessarily  produce  a  change  m  the  value  of  y . 

4.  The  difference  between  any  two  values  of  a  variable  is 
called  an  increment  of  the  variable,  since  it  may  be  regarded  as 
the  amount  that  must  be  added  to  the  first  value  to  produce  the 
second.  An  increment  is  denoted  by  writing  the  letter  J  before 
the  variable  in  question.  Thus  the  difference  between  two  val- 
ues of  a  variable  x  would  be  written  Ax^  A  being  merely  a  sjai- 


Chap.  I.]  INTRODUCTION.  •         3 

bol  for  the  word  increment^  and  the  expression  Ax  representing 
a  single  quantity.  It  is  to  be  noted  that  as  an  increment  is  a 
difference,  it  may  be  either  positive  or  negative. 

5.  If  a  variable  which  changes  its  value  according  to  some 
law  can  he  made  to  approach  some  fixed ^  constant  value  as 
nearly  as  lue  please,  but  can  never  become  equal  to  it,  the  con- 
stant is  called  the  li7nit  of  the  variable  under  the  circumstances 
in  question. 

6.  For  example  ;  the  limit  of  -,  as  n  increases  indefinitely,  is 

n 

zero ;  for  by  making  n  sufficiently  great  we  can  evidently  de- 
crease -  at  pleasure,  but  we  can  never  make  it  absolutely  zero. 
n 

The  sum  of  n  terms  of  the  geometrical  progression  1,  ^,  ^,  ^, 
&c.,  is  a  variable  that  changes  as  n  changes,  and  if  n  is  in- 
creased at  pleasure,  the  sum  will  have  2  for  its  limit ;  for,  by  the 
formula  for  the  sum  of  a  geometrical  progression, 

ar**  —  a 


In  this  case, 


s  = , 

r-1 

-1-1      1- 

2" 

1 

i-1     .i 

2  2 


1 


By  increasing  n,  —  can  be  made  as  small  as  we  please,  but  can- 
not become  absolutely  zero  ;  the  numerator  can  then  be  made  to 
approach  the  value  1  as  nearly  as  we  please,  and  the  limit  of  the 
value  of  the  fraction  is  obviously  2. 

We  say,  then,  that  the  limit  of  the  sum  of  n  terms  of  the  pro- 
gression 1,  ^,  I,  ^,  &c.,  as  n  increases  indefinitely,  is  2. 

In  both  of  these  examples  the  variable  increases  towards  its 
limit,  but  remains  always  less  than  its  limit.  This,  however,  is 
not  always  the  case.  The  variable  may  decrease  towards  its 
limit  remaining  always  greater  than  the  limit,  or  in  approach- 
ing its  limit,  it  may  be  sometimes  greater  and  sometimes  less 


4         •  DIFFEEENTIAL  CALCULUS.  [Art.  7. 

than  the  limit.  Take,  for  example,  the  sum  of  n  terms  of  the 
progression  1,  —  |-,  i,  —  i,  tf»  <^c.,  where  the  ratio  is  —  ^. 
Here  the  limit  of  the  sum  as  n  increases  indefinitely^  is  f .  Let 
n  start  with  the  value  1  and  increase  ;  when 

and  is  greater  than  the  limit  f  ;  when 

and  is  less  than  |,  but  is  nearer  f  than  1  was  ;  when 

and  is  greater  than  f  ;  when 

71  =  4,  s  =  |, 

and  is  less  than  f ;  and  as  n  increases,  the  values  of  s  are  alter- 
nately greater  and  less  than  the  limit  f ,  but  each  value  of  s  is 
nearer  f  than  the  value  before  it. 

7.  It  follows  immediately  from  the  definition  of  a  limits  that  the 
difference  between  a  variable  and  its  limit  is  itself  a  variable 
which  has  zero  for  its  limits  and  in  order  to  prove  that  a  given 
constant  is  the  limit  of  a  particular  variable,  it  will  alwaj^s  suf- 
fice to  show  that  the  diflference  between  the  two  has  the  limit 
zero. 

For  example ;  it  is  shown  in  elementary  geometry  that  the 
difference  between  the  area  of  any  circle  and  the  area  of  the 
inscribed  or  circumscribed  regular  polygon  can  be  made  as  small 
as  we  please  by  increasing  the  number  of  sides  of  the  polygon, 
and  this  diflference  evidently  can  never  become  absolutely  zero. 
The  area  of  a  circle  is  then  the  limit  of  the  area  of  the  regular 
inscribed  or  circumscribed  pol3'gon  as  the  number  of  sides  of  the 
polygon  is  indefinitel}^  increased. 

It  is  also  shown  in  geometry,  that  the  difference  between  the 
length  of  the  circumference  of  a  circle  and  the  length  of  the 
perimeter  of  the  regular  inscribed  or  circumscribed  polygon  can 
be  decreased  indefinitely  by  increasing  at  pleasure  the  number 
of  sides  of  the  polygon,  and  this  diflference  evidently  can  never 


Chap.  I.]  INTEODUCTION.  6 

become  zero.  The  length  of  the  circumference  of  a  circle  is 
then  the  limit  of  the  length  of  the  perimeter  of  the  regular  in- 
scribed or  circumscribed  polygon  as  the  number  of  sides  of  the 
latter  is  indefinitely  increased. 

8.  The  fundamental  proposition  in  the  theory  of  limits  is  the 
following 

Theorem.  — If  two  variables  are  so  related  that  as  they  change 
they  keep  always  equal  to  each  other ^  and  each  approaches  a  limit, 
their  limits  are  absolutely  equal. 

For  two  variables  so  related  that  they  are  always  equal  form 
but  a  single  varying  value,  as  at  any  instant  of  their  change 
they  are  by  hypothesis  absolutely  the  same.  A  single  varjdng 
value  cannot  be  made  to  approach  at  the  same  time  two  different 
constant  values  as  nearly  as  we  please  ;  for,  if  it  could,  it  could 
eventuall}'  be  made  to  assume  a  value  between  the  two  constants  ; 
and,  after  that,  in  approaching  one  it  would  recede  from  the 
other. 

9.  As  an  example  of  the  use  of  this  principle,  let  us  prove 
that  the  area  of  a  circle  is  one-half  the  product  of  the  length  of 
its  radius  b}'  the  length  of  its  circumference. 

Circumscribe  about  the  circle  any  regular  polygon,  and  join  its 
vertices  with  the  centre  of  the  circle,  thus  divid- 
ing it  into  a  set  of  triangles,  each  having  for  its 
base  a  side  of  the  polygon,  and  for  its  altitude 
the  radius  of  the  circle.  The  area  of  each 
triangle  is  one-half  the  product  of  its  base  by  the 
radius.  The  sum  of  these  areas,  or  the  area  of 
the  polygon,  is  one-half  the  length  of  the  radius 
by  the  sum  of  the  lengths  of  the  sides,  that  is,  by  the  length 
of  the  perimeter  of  the  polygon.  If  J.'  is  the  area,  and  P  the 
perimeter  of  the  polygon,  and  R  the  radius  of  the  circle,  we 
have 

A=^RP; 

a  relation  that  holds,  no  matter  what  the  number  of  sides  of 


6  DIFFERENTIAL  CALCULUS.  [Art.  10. 

the  polygon.  A'  and  ^BP  evidently  change  as  we  change  the 
number  of  sides  of  the  polygon ;  the}^  are  then  two  variables  so 
related  that,  as  they  change,  they  keep  always  equal  to  each 
other.  As  the  number  of  sides  of  the  potygon  is  indefinitely 
increased.  A'  has  the  area  of  the  circle  as  its  limit ;  P  has  the 
circumference  of  the  circle  as  its  limit.  Let  A  be  the  area  and 
C  the  circumference  of  the  circle  ;  then  the 

limit  A'=A, 

and  the  limit  ^RP=^  EC. 

By  the  Theorem  of  Limits  these  limits  must  be  absolutely  equal ; 

.'.  A  =  }EC.  Q.E.D. 

10.  It  is  of  the  utmost  importance  that  the  student  should 
have  a  perfectly  clear  idea  of  a  limit,  as  it  is  by  the  aid  of  this 
idea  that  many  of  the  fundamental  conceptions  of  mechanics 
and  geometry  can  be  most  clearly  realized  in  thought. 

11.  Let  us  consider  briefly  the  subject  of  the  velocit}^  of  a 
moving  body. 

The  mean  velocity  of  a  moving  bod}^,  during  an}^  period  of 
time  considered,  is  the  quotient  obtained  by  dividing  the  dis- 
tance traversed  by  the  body  in  the  given  period  by  the  length 
of  the  period,  the  distance  being  expressed  in  terms  of  a  unit 
of  length,  and  the  length  of  the  period  in  terms  of  some  unit  of 
time. 

If,  for  example,  a  body  travels  60  feet  in  3  seconds,  its  mean 
velocity  during  that  period  is  said  to  be  Af ,  or  20  ;  and  the 
body  is  said  to  move  at  the  mean  rate  of  20  feet  per  second. 

The  velocity  of  a  moving  body  is  uniform  when  its  mean  ve- 
locity is  the  same  whatever  the  length  of  the  period  considered. 

The  actual  velocity  of  a  moving  body  at  any  instant,  is  the 
limit  which  the  body's  meaii  velocity  during  the  period  imme- 
diately  succeeding  the  instant  in  question  approaches  as  the 
length  of  the  period  is  indefinitely  decreased.     In  the  case  of 


Chap.  I.]  INTRODUCTION.  T 

uniform  velocity,  the  actual  velocity  at  any  instant  is  obviously 
the  same  as  the  actual  velocity  at  any  other  instant. 

If  the  actual  velocit}^  of  a  moving  body  is  continually  changing, 
the  body  is  said  to  move  with  a  variable  velocity. 

12.  If  the  law  governing  the  motion  of  a  moving  body  can 
be  formulated  so  as  to  express  the  distance  traversed  by  the 
body  in  any  given  time  as  a  function  of  the  time,  we  can  indicate 
the  actual  velocity  at  any  instant  very  simply  by  the  aid  of  the 
increment  notation  already  explained.  Represent  the  distance 
by  s  and  the  time  by  t.     Then  we  have 

s=ft. 

Suppose  we  want  to  find  the  actual  velocity  at  the  end  of  t^ 
seconds.  Let  M  be  any  arbitrary'  period  immediately  succeeding 
the  end  of  ?o  seconds  (it  can  fairly  be  considered  an  increment 
as  we  reall}'  increase  the  time  during  which  the  body  is  sup- 
posed to  have  moved  by  At  seconds) ,  and  let  As  be  the  distance 
traversed  in  that  period.     Then,  by  definition,  the  mean  velocity 

As 
during  the  period  At  is  — ,  and  the  actual  velocit}'  desired  is  the 

limit  approached  b}^  this  ratio  as  J^  approaches  zero.  We  shall 
indicate  this  by 

limit  r^l 

At=0\_AtJ 

which  is  to  be  read  "the  limit  of  J. 9  divided  by  Jf,  as  At  ap- 
proaches zero'' ;  the  sign  =  standing  for  the  word  approaches. 

13.  Take  a  numerical  example.  In  the  case  of  a  body  falling 
freely  in  a  vacuum  near  the  surface  of  the  earth,  the  relation 
connecting  the  distance  fallen  with  the  time  is  nearly 


s  being  expressed  in  feet  and  t  in  seconds ;  required,  the  actual 
velocity  of  a  falling  body  at  the  end  of  t^  seconds.  Let  At  seconds 
be  an  arbitrary  period  immediately  after  the  end  of  tQ  seconds, 


8  DIFFERENTIAI.    CALCULUS.  [A-RT  14. 

then  in  ^o  4-  ^i  seconds 

the  body  would  fall  16  (^o  +  ^O'  feet, 

or  16  ^o'  + 32^0^^  +  16(^0"  feet. 

In  to  seconds  it  falls  16^o^  feet,  so  that  in  the  period  Jt  in  question, 
it  would  fall         16 ^o'  4-  32 ^0 ^«  +  16  {My-16to'  feet, 

or  32^0^^+16(^0''  feet, 

which  must  therefore  be  Js.   If  Vq  be  the  required  actual  velocity, 

limit  \  As~]  -,      A  u.    -lo 

''«  =  ^«=oL^J'  by  Art.  12. 

and  obviously  /l/^^0    j7    =32^o- 

Hence  '^o  =  32  to, 

the  result  required  ;  and  in  general,  the  velocity  v  at  the  end  of  t 
seconds  is 

-^  =  32^. 

14.  Let  us  now  consider  a  geometrical  problem  :  To  find  the 
direction  of  the  tangent  at  any  point  of  a  given  curve. 

The  tangent  to  a  curve,  at  any  given  point,  is  the  line  with 
which  the  secant  through  the  given  point  and  any  second  point 
of  the  curve,  tends  to  coincide  as  the  second  point  is  brought 
indefinitely  near  the  first.  In  other  words,  its  position  is  the 
limiting  position  of  the  secant  line  as  the  second  point  of  inter- 
section approaches  the  first,  i.e.,  a  position  that  the  secant  line 
can  be  made  to  approach  as  nearly  as  we  please,  but  cannot 
actually  assume. 

15.  Suppose  we  have  the  equation  of  a  curve  in  rectangular 
coordinates,  and  wish  to  find  the  angle  r  that  the  tangent  at  a 
given  point  {xQ^y^)  of  the  curve  makes  with  the  axis  of  X ;  that 
is,  what  is  called  the  inclination  of  the  curve  to  the  axis  of  X. 


Chap.  I.] 


INTRODUCTION. 


9 


The  equation  of  the  curve  enables  us  to  express  y  in  terms  of 
ic,  that  is,  as  a  function  of  x.     We  have  then 


Let  Xq-\-Ax 

be  the  abscissa  of  any  second  point  P  of  the  curve,  and 

the  corresponding  ordinate.     If  (p  is  the  angle  which  the  secant 
through  Pq  and  P  makes  with  the  axis  of  X,  it  is  clear  from  the 


figure  that 


tan  (f  — 


Ax' 


As  P  approaches  Pq,  that  is,  as  Ax  decreases  toward  zero,  <p 
evidently  approaches  r  as  its  limit,  and  tan  (p  of  course  ap- 
proaches indefinitely  tanr.  Hence,  by  the  fundamental  theo- 
rem of  limits  (Art.  8), 


tanr 


imit   \^ 


limit 

Ax 


16.   Take  a  particular  example.     To  find  the  inclination  r^  to 
the  axis  of  X,  of  the  parabola 

?/"  =  2  mx 

at  the  point  {x^^y^  of  the  curve. 

If  the  abscissa  of  P  is  a^o  +  ^^5  its  ordinate  yQ-\-Ay  must  be 

^l2m{xQ  +  Ax)2, 

as  is  clear  from  the  equation  of  the  curve,  which  may  be  written 

y  =  ^(^2mx). 


10  DrFFERENTLA^L  CALCULUS.  [Art.  17. 

2/o  =  V(2wia;o), 

Ay  must  be  yj[2m  {xq  +  Ax) ]  —  ^{2mXo) . 

Ay  _  V[2m  (xq-{-Ax)']  —  ^(2mxQ)  . 
Ax~  Ax 

or,  multiplying  numerator  and  denominator  by 

V[2m(aJo  +  ^a;)]  +  -^{^mx^) 

to  rationalize  the  numerator, 

Ay  _  2m{xQ-\- Ax) —  ^mxQ 

Ax''  Ax  j^[2m  {xQ-\-Ax)~\-\-'sJ{2mxQ)\^ 

and  tan  To:     ''-''  ^^^"^  ^^ 


^  limit  ri^~l  = 
Ja;=OLJicJ 


2^(2mxQ)       ^{2mXo)       y^ 
At  any  point  (a;,y)  of  the  parabola  we  should  have 


,  m 

tan  r  = — 

y 


At  the  extremity  of  the  latus  rectum,  i.e.,  at  the  point  j  — ,  m  J, 

tan  r  =  —  =  1 , 
m 

and  r  =  45°, 

a  familiar  property  of  the  parabola. 

17.  Each  of  the  problems  we  have,  just  considered  has  re- 
quired for  its  solution  the  investigation  of  the  limit  approached 
by  the  ratio  of  corresponding  increments  of  a  function  and  of 
the  variable  on  which  it  depends,  as  the  increment  of  the  inde- 
pendent variable  approaches  zero.  Such  a  limit  is  called  a  de- 
rivative, or  a  differential  coefficient,  and  the  study  of  its  form 
and  properties  is  the  fundamental  object  of  the  Differential  Cal- 
culus, 


Chap.  II.]  DIFFERENTIATIOK.  11 


CHAPTER  II. 

DIFFERENTIATION  OF   ALGEBRAIC    FUNCTIONS. 

18.  If  2/  is  a  function  of  ic,  the  limit  of  the  ratio  of  an  incre- 
Inent  of  y  to  the  corresponding  increment  of  x,  as  the  increment 
ofx  approaches  zero^  is  called  the  derivative  of  j  with  respect  to 
X,  and  is  indicated  by  D^y,  D^  being  merely  an  abbreviation  for 
derivative  with  respect  to  x.*  For  an}^  particular  value  of  ic,  this 
limit,  as  we  shall  see,  will,  in  general,  have  a  perfectly  definite 
value ;  but  it  will  change  in  value  as  x  changes ;  that  is,  the 
derivative  will,  in  general,  be  a  new  function  of  x. 

Since  our  definition  of  derivative  requires  that  y  should  be  a 
function  of  x,  that  is,  should  change  when  x  changes,  it  follows 
that  a  constant  can  have  no  derivative ;  and  if  we  attempt  to 
find  the  derivative  of  a  constant  by  the  method  which  we  should 
use  if  it  were  a  function  of  x^  we  shall  be  led  to  this  same  con- 
clusion. Let  a  be  any  constant ;  then  the  increment  produced 
in  a,  by  giving  x  any  increment,  is  absolutely  0 ;  the  ratio  of 
this  increment  to  the  increment  of  x  must  then  be  0  ;  and  as  this 
ratio  is  alwaj's  0,  its  limit,  when  we  suppose  the  increment  of  x 
to  decrease,  must  be  0.     Therefore 

A«  =  0.  [1] 

19.  The  general  method  of  finding  the  derivative  of  any  given 
function  o/x,  is  immediately  suggested  b}^  the  definition  of  a  de- 
rivative. Take  two  values  of  x,  Xq  and  Xq  -f  Jx,  and  find  the 
corresponding  values  of  the  given  function ;  the  diflference  be- 
tween them  is  obviously  the  increment  of  the  function,  corre- 
sponding to  the  increment  ^x  of  x.     The  limit  of  the  ratio  of  the 

*  The  names  differential  coefficient  and  derived  function,  and  the  notation  ^  in 
place  of  Dxy,  are  also  in  common  use 


12  DIFFERENTIAL   CALCULUS.  [Art.  20. 

two  increments,  as  Jx  approaches  zero,  will  be  the  value  of  the 
derivative  for  the  particular  value  x^  of  a,  and  we  may  indicate 
it  by  \_Dxy~\x=XQ'  As  Xq  was  taken  at  the  start  as  any  value 
of  ic,  the  subscripts  may  be  dropped  in  the  result,  and  the  de- 
rivative will  then  be  expressed  as  an  ordinary  function  of  x. 
The  method  may  be  formulated  as  follows :  — 


[A/.].=.  =  «™t, 


y{Xo-^Jx)-fXo~.  j-j-j 

uX 


'] 


The  student  will  observe  that,  in  the  problems  in  Arts.  13  and  16, 
we  have  really  found  Dt(Wf)  and  D^  {\/2mx)  by  the  method 
just  described. 

Examples. 
Find 

{l)D^(20x);    (2)A(a^);     (3)  aQ  ;    (4)i).(V^); 
by  the  general  method. 

^ns.  (1)20;    (2)3a^;    (3)-^-;    (4)        ^ 


x'       '  '  2  V(^) 

20.  In  order  to  deal  readily  with  problems  into  which  deriva- 
tives  enter,  it  is  desirable  to  work  out  a  complete  set  of  formu- 
las, or  rules  for  finding  the  derivatives  of  ordinar}'  functions ; 
and  it  will  be  well  to  begin  by  roughly  classifying  functions. 

The  functions  ordinarily  considered  are  :  — 

(1)  Algebraic  Functions :  those  in  which  the  onl}' operations 
performed  upon  the  variable,  are  the  ordinar}^  algebraic  opera- 
tions, namely :  Addition,  Subtraction,  Multiplication,  Division, 
Involution,  and  Evolution. 

Example.  ^x^  +  ^^(x  —  l). 

(2)  Logarithmic  Functions:  those  involving  a  logarithm  of 
the  variable,  or  of  a  function  of  the  variable. 

Examples.  x  log  x ; 

log  (x^  —  ax-\-b). 


Chap.  II.]  DIFFERENTIATION.  13 

(3)  Exponential  Functions:  those  in  which  the  variable,  or 
a  function  of  the  variable,  appears  as  an  exponent. 

Example.  a^(*^-^). 

(4)  Trigonometric  Functions : 
Example.  cos  a?  —  sin^  ic. 

21.   We  shall  consider  first,  the  differentiation*  of  Algebraic 
Functions  of  x. 

Required  D^  (ax)  where  a  is  a  constant. 
By  the  general  method  (Art.  19), 

rn      T  -  liinit  [a{xQ-\-Ax)  -ax;\_  limit  VoJx\ 

l^^^^\x=:x^- ^x=0\_  Ax  J     Jaj=0|_Ja;J 

limit    r  -1 

.-.  D^{ax)=a.  [1] 

If  a  =  1 ,  this  becomes  D^x  =  \.  [2] 

Required  D^x"  where  n  is  a  positive  integer. 

rn    nn         _  limit  [{x,  +  AxY-x,--\ 
l^^^  \x=xq- _\x^0\_  Jx  J* 

By  the  Binomial  Theorem, 

(X0+  Jxy=  XfT  +  nX(r-''AX-{.  '^(^-'^)  x^n-2  (  J^)2  _f_  _  _f_  (  J^,^n 


uX  Z 

Each  term  after  the  first  contains  Ax  as  a  factor,  and  therefore 
has  zero  for  its  limit  as  Ax  approaches  zero,  so  that 

limit   r(a^o+Ja^)--^o-1_        ._,. 
Ax=^\_  Ax  J"""^*^       ' 

*  To  differentiate  is  to  find  the  derivative. 


14  DIFFERENTIAL  CALCULUS.  [Art.  22. 

As  Xq  is  any  value  of  x,  we  may  drop  the  subscript,  and  we  have 
D^x''  =  nx''-\  [3] 

22.  We  shall  next  consider  complex  functions  composed  of 
two  or  more  functions  connected  by  algebraic  operations ;  the 
sum  of  several  functions,  the  product  of  functions,  the  quotient 
of  functions. 

Required  the  derivative  of  u-\-v  -\-w^ 

where  each  of  the  quantities  it,  v,  and  w  is  a  function  of  x. 

Let  ^x  be  any  increment  given  to  x,  and  Au^  Jv,  and  Aw  the 
corresponding  increments  of  w,  v^  and  w.  Then,  obviously,  the 
increment  of  the  sum  w  +  v  +  w 

is  equal  to  Au-\-Av-\-  Aw,  and  we  have 

D(u^v-^w)  =  liniit  rA_u±Av±Awl^  limit  f^!*  .  ^  ,  ^1 
^  ^      Ax=0\_  Ax  J     Ax=0\_Ax^Ax^Axj 

^  limit  fj^l  ,    limit  r^l,    limit  f^l. 
Ax=0  [_AxyAxd=0  \_AxyAx=0  \_Axy 

but,  since  Au  and  Ja?  are  corresponding  increments  of  the  func- 
tion u  and  the  independent  variable  x, 

limit  ff^l^^ 

.    vi  limit  r^'^l      ^ 

m  hke  manner  ^^^^^  |^-J  =  A^, 

or,^  limit  r^H      ^ 

Ax=0  \_AxJ        '     ' 

hence  D^{u  +  v+w)=D^u-\-D,v  +  D^w.  [1] 

It  is  easily  seen  that  the  same  proof  in  effect  may  be  given, 
whatever  the  number  of  terms  in  the  sum,  and  whether  the 
connecting  signs  are  plus  or  minus.  So,  using  sum  in  the  sense 
of  algebraic  sum,  we  can  say.  the  derivative  with  respect  to  x 


Chap.  II.] 


DIFFERENTIATION. 


15 


of  the  sum  of  a  set  of  functions  of  x  is  equal  to  the  sum  of  the 
derivatives  of  the  separate  functions. 

23.  Required^  the  derivative- of  the  product  uv,  where  u  and  v 
are  functions  of  x. 

Let  a^o,  Uq^  and  Vq  be  corresponding  values  of  a;,  u^  and  v  ;  let 
Ax  be  an  increment  given  to  a;,  and  Au  and  Av  the  corresponding 
increments  of  u  and  v.     Then, 

(Wo  +  ^U)  (Vo  -f-  ^V)  —  Uq  Vq 


lD^{uv)^,=,,^ 


limit 

Ax=0 


Ax 


■} 


{uq-{-  Au)  (vq-\-  Av)  —  UqVq  =  UqAv  -\-  Vq  Au  +  Au  Av 

-,          rn  /     M          -  limit   [u^Av +  VqAu-{-AuAv~\ 
and         [A0^^)].=.,- j^^O  L Tx J 


+ 


hmit 
Ja;=0 


_  limit   L^]^lmiit   L 

ilq  does  not  change  as  Ax  changes,  and 

limit  r^1_  1-2)^-1 
Ax=oIaxJ~^    ''^'=^0^ 


and  in  like  manner 


imit   L  ^^0_      rn     i 


limit 

Ja; 


may  be  written  Au  —  or  Jv  — .     Let  us  consider 

Ax  Ax  Ax 


limit 

Ax^O 


Au 


Avl 
Ax] 


As  Ax  approaches  0,  Au,  being  the  corresponding  increment  of 

the  function  w,  will  also  approach  0  ;    and  the  product  Au  — 

Ax 

will  approach  0  as  its  hmit,  if  —  approaches  any  definite  value ; 


16  DIFFERENTIAL   CALCULUS.  [Art.     23. 

that  is,  if  D^v  has  a  definite  value.     It  is,  however,  perfectly 

conceivable  that  —  may  increase  indefinitelv  as  Ax  approaches 

Ax 

Av 
zero,  instead  of  having  a  definite  limit ;  and,  in  that  case,  if  — - 

Ax 

should  increase  rapidly  enough  to  make  up  for  the  simultaneous 

Av 
decrease  in  Jw,  the  product  Au  —  would  not  approach  zero. 

Ax 

We  shall  see,  however,  as  we  investigate  all  ordinary  functions, 
that  their  derivatives  have  in  general  fixed  definite  values  for 
any  given  value  of  the  independent  variable ;  but,  until  this  is 
established,  we  can  only  say,  that 


!=0 1_       Ax] 


Hmit  ,  ^^  _  .      Q 
Ax^C  '  --  I      ^ 


Av       Au 
when  —  or  —  has  a  definite  limit,  as  Ax==0  ;  that  is,  when 
Ax       Ax 

has  a  definite  value.     With  this  proviso,  we  can  say, 

or,  dropping  subscripts, 

Dj.  (uv)  =  uD^v  -\-  vD^u.  [1] 

Divide  through  by  uv^  and  we  have  the  equivalent  form, 

D,{uv)  ^D,u     D^v  p^-j 

UV         ~      U  V 

If  we  have  a  product  of  three  factors,  as  uvw^  we  can  repre- 
sent the  product  of  two  of  them,  say  vw^  by  z,  and  we  have 

D^  {uvw)  _  D^  {iLz)      D^u     D^z 
uvw      ~      uz      ~  ~u     '     z~ ' 

But  D^z^D,(vw)  ^D,v     D^w^ 

Z  VW  V  w     ^ 


Chap.  II.] 


DIFFERENTIATION. 


17 


D^(uvw)     D^u     D^v     D^w 


[3] 


This  process  may  be  extended  to  any  number  of  factors,  and  we 
shall  have  the  derivative  of  a  x>roduct  of  functions  divided  by  the 
product  equal  to  the  sum  of  the  terms  obtained  by  dividing  the 
derivative  of  each  function  by  the  function  itself. 


24.    Required  the  derivative  of  the  quotient  -^  where  u  and  v 

V 

are  functions  of  a;.     Emplo3lng  our  usual  notation,  we  have 


A 


_  limit 


"wo  +  ^u 

Vq-\-  Av 

Ax 

but 


Uo-{- Au      Uq_VqAu  —  UqAv 
Vq  -j-  Av      Vq       v^  +VqAv 


and  dividing  b}^  Jx,  we  have 


D. 


limit 
Jx=0 


Au         Av 
Ax         Ax 


^'o[A^^]^  =  .-^^o[A^],:=^, 


and  dropping  subscripts, 


^•fi^= 


vD^u  —  uD^v 


[1] 


Examples. 


Find 


(1)  A[a^  +  a^-VW];  (2)  A[a^V(^)];  (3)I>«^. 

Ans. 
(1)  3a^+l ?—  ;    (2)^-+2a;VW;    (3) :^ 


18 


DIFFERENTIAL  CALCULUS. 

r 


[Art.  25 


(4)  Find,bjArt.  24,  [l],i>. 


(5)  Deduce  D^x"*  from  last  i^art  of  Art.  23. 

25.  If  the  quantity  to  be  differentiated  is  a  function  of  a 
function  of  a?,  it  is  alwa3's  theoreticailj'  possible,  by  performing 
the  indicated  operations,  to  express  it  directh'  as  a  function  of 
x^  and  then  to  find  its  derivative  by  the  ordinary  rules ;  but  it 
can  usually  be  more  easil3'  treated  by  the  aid  of  a  formula  which 
we  shall  proceed  to  estabhsh. 

Required^  D^fy,  y  being  itself  a  function  ofs..  Let  Xq  and  ?/o 
be  corresponding  values  of  x  and  y ;  let  Ax  be  an}-  increment 
given  to  x,  and  Ay  the  corresponding  increment  of  y  ;  then 


[^x/2/].=.o     Ax 
and  this  can  be  written 


Kmit  r/(yo  +  4v)  -/yo"| 

^x=0[_  Ax  \ 


limit 
Ax=0 


'f{yo+^y)-fyo  Ay 


] 


As  Ax  and  Ay  are  corresponding  increments,  they  approach  zero 
together ;  hence 


is  the  same  as 


limit 
limit 


,f(yo±_Ay)_-fy 

Ay 


'f(yo-hAy)-fy 
Ay 


which  is  equal  to  lD^fy']y=y^' 

or,  dropping  subscripts, 

DJy==Dyfy-D,y-  [i] 

This  gives  immediately,  as  extensions  of  Art.  21,  [1]  and  [3], 
Dx  (ay)  =  aD^y,  D^y'^  =  ny^-'^D^y. 


Chap.  II.]  DIFFERENTIATION.  19 

26.  Art.  21,  [3]  can  now  be  readily  extended  to  the  case 
where  n  is  any  numher  positive  or  negative^  whole  or  fractional. 

Let  n  be  a  negative  whole  number  —m,m  of  course  being  a 
positive  whole  number.     Let 

2/  =  X"  =  x~"*, 

then  we  want  D^y.     Multiplying  both  members  of 

y  =  x~"' 

by  x^,  we  have  x^'y^  1. 

Since  x'^y  is  a  constant,  its  derivative  with  respect  to  x  must  be 
zero ;  but  by  Art.  23,  [1]  and  Art.  21,  [3], 

A  [^"^2/]  =x"'D^y-\-  yD^x"^  =  x"'D^y-^  mx'^~'^y 

m  being  a  positive  integer  ; 

.*.  x"'D^y-\-mx"'~'^y=0, 

and  D^y=—mx~^y=—mx~"'~^  =  nx'^~^,  q.e.d. 

Let  n  be  any  fraction  -  where  p  and  q  are  integers  either  posi- 


tive  or  negative. 

As  before,  let 

2/  =  aj"  =  a;? ; 

required  D^y. 

Clearing 

y  =  x'^ 

of  radicals, 

we  have 

y^=xP', 

and  since  the  two  members  are  equal  functions  of  x,  their  deriva- 
tives must  be  equal ; 

D,y^  =  D,xP, 

or  qy''-^D^y=pxP-\ 

p  xP~^     p     xP~'^       p     |-i         ^__^ 
and  ■^'y~a'  ir^^^~a       i      \~'q'^^     =  wa;"    .       q.e.d, 


20  DIFFERENTIAL  CALCULUS.  [Art.  26. 

The  formula^  D^  a;"  =  n»"- % 

Art.  21,  [3],  holds,  then,  whatever  the  value  of  «. 

Example. 

Prove  Art.  24,  [1]  by  the  aid  of  Art.  21,  [3]  and  Art.  23, 
[1],  regarding  -  as  a  product,  namely  uv~^. 

By  the  aid  of  these  formulas, 

Z).a  =  0;  [1] 

D^ax  =  a\  [2] 

Z>.x=l;  [3] 

Z>^a;"=wa;^-^'  [4] 

D,{u-\rV-\-w)  =D,u~\-D,v  +D,w ;  [5] 

D^  {uv)  =uD,v-\-vD,u]  ■                                     [6] 


D. 

K^)- 

D. 

(fy)  = 

--D,{fy). 

A2/; 

[8] 

any  algebraic  function,  no  matter  how  complicated,  may  be  dif- 
ferentiated. 


Examples. 
Find  D^  u  in  each  of  the  following  cases  :  — 

(1)  w  =  m  +  wfl7.  A71S.   D^u  =  7i. 

(2)  w  =  (a  -f  bx)  a^.  Ans,   D^u=  (Ux  +  Sa)  a?. 


Chap.  II.  1  DIFFERENTIATIOK.  21 

(3)    u=^{x^-\-a^). 

Solution :  u  =  ^J  (oc^  +  a^)  =  (jx^ -\-  a^  i. 

Let  y  =  x^-{-a^, 

then  u  =  y^. 

D,u  =  D^yh  =  D,yh . D,y     by  Art.  26,  [8], 
Dyy'^  =  ^y-'^  byArt.  26,  [4J, 

D,y  =  2x\ 


(^>  ^  =  (iT-J- 


V(^  +  aO 


Ans.  D^u= 


naf~^ 


(l+a;)"+i 


(a  -i-a^)2  (a  +  iT)^ 

(6)    w=  (l  +  aj)V(l-^)-  ^s.  2),w=     1-^^ 


2V(l-aj) 


2(H-^)V(»-a^) 


(9)    «=J(|^)-  ■4»«-  ■D.«=  ^ 


(l_a;)V(l-»2) 


(l«)"-V(l+^)_V(l-a^)- 


^«..I>.«=-|[l+^^]. 


22 


DIFFERENTIAL   CALCULUS. 


[Art.  27. 


CHAPTER  III. 

APPLICATIONS. 

Tangents  and  Normals, 

27.   "We  have  shown,  in  Art.  15,  that  the  angle  r,  made  with 
the  axis  of  oj  by  the  tangent  at  any  given  point  of  a  plane  ciure, 


when  the  equation 


=fx 


of  the  curve,  referred  to  rectangular  axes,  is  known,  maybe  found 
by  the  relation 


tanr 


^  limit  r^l 


where  Ay  and  Ax  ar^  corresponding  increments  of  y  and  a;,  the 
coordinates  of  a  point  of  the  curve.  If  the  point  be  {xQ^y^) ,  we 
have,  then, 

tanro  =  [Z),2/]x=^„- 


At  any  point  {x^y)  tan  r  =  D^y. 

T 


[1] 


A  line  perpendicular  to  any  tangent,  and  passing  through  the 
point  of  contact  of  the  tangent  with  the  curve,  is  called  the  noy^- 
mal  to  the  curve  at  that  point.     If  Vq  be  the  angle  which  the 


Chap.  III.]  APPLICATIONS.  23 

normal  at  the  point  (aJo?2/o)  makes  with  the  axis  of  X,  then  it  is 
evident  from  the  figure,  that 

and  from  trigonometry, 

1 


tan  Va  =  —  cot  To  = 


Of  course,  for  any  point  {x,y) 

tanj'=--f-.  m 

28.  Since  the  tangent  at  (o^o^^/o)  passes  through  (xq^i/q)^  and 
makes  an  angle  tq  with  the  axis  of  x,  its  equation  will  be,  by 
analytic  geometry, 

y-yi)  =  tan  To  (x  -  Xo)  ; 
or,  since  tan  tq  =  [A2/]x=^o' 

y-yo=  [A2/]^=aro(^  -  ^o) .  [1] 

In  like  manner,  the  equation  of  the  normal  at  (a!o,?/o)  is  found  to 
be  y-y^=-——- (x-xo).  [2] 

L-^xyjx=XQ 

The  distance  from  the  point  of  intersection  of  the  tangent 
with  the  axis  of  X  to  the  foot  of  the  ordinate  of  the  point  of 
contact,  is  called  the  subtangent,  and  is  denoted  by  t^. 

The  distance  from  the  foot  of  the  ordinate  of  the  point  to  the 
intersection  of  the  normal  with  the  axis  of  X,  is  called  the  sub' 
normal,  and  is  denoted  by  n^. 

In  the  figure,  TA  and  AN  are  respectively  the  subtangent  an(!i 
subnormal,  corresponding  to  the  point  (xo,?/o)  of  the  curve. 

Obviously  |«  =  tan  tq  =  [A2/]a:=x-o' 


24  DIFFEKENTIAL  CALCULUS.  [Art.  28 


VO         .  /.o.^o  X  .  1 


and         £l  =  tan  (180°  -vq)=-  tan  v^ 


hence  for  the  point  (a;o,2/o) » 


umz:,  '  »'=^»t^-^^^=v 


The  distance  from  the  intersection  of  the  tangent  with  the 
axis  of  X  to  the  point  of  contact  is  sometimes  called  the  length 
of  the  tangent^  and  may  be  denoted  by  t. 

The  distance  from  the  point  at  which  the  normal  is  drawn  to 
the  point  where  the  normal  crosses  the  axis  of  X  is  sometimes 
called  the  length  of  the  normal^  and  may  be  denoted  by  n. 

It  is  easily  seen  from  the  figure,  that 

and  w=VW  +  Wx'); 

hence  t  =  y,  ^D^y']-^^^  V(l  +  \.D,y-]l^^^ , 

and  ^  =  2/oV(l+[A2/]|=.«). 

For  any  point  {x^y) ,  our  formulas  become 

^^  =  7r-5  [3] 

'^x  =  yD,y',  [4] 

^=2/[^.2/rV(n-[^.2/?);  [6] 

^  =  2/V(l+[A2/?).  [6] 

Examples. 

(1)    Show  that  the  inclination  of  a  straight  line  to  the  axis  of 
X  is  the  same  at  every  point  of  the  Une  ;  i.e. ,  prove  tan  r  constant. 


Chap.  III.]  APPLICATIONS.  25 

(2)  Show  that  the  subnormal  in  a  parabola 

2/2  =  2  ma; 

is  constant,  and  that  the  subtangent  is  alwaj^s  tvrice  the  abscissa 
of  the  point  of  contact  of  the  tangent. 

(3)  Find  what  point  of  the  parabola  must  be  taken  in  order 
that  the  inclination  of  the  tangent  to  the  axis  of  X  may  be  45°. 

29.   If  the  equation  of  the  curve  cannot  be  readily  thrown  into 
the  form  y—fx^ 

D^y  may  be  found  by  differentiating  both  members  with  respect  to 
X  and  solving  the  resulting  equation  algebraically^  regarding  D^y 
as  the  unknown  quantity. 

For  example  ;  required  the  equation  of  the  tangent  to  a  circle 
at  the  point  (a;o,2/o)  of  the  curve.     The  equation  of  a  circle  is 

r  being  constant.  Differentiating  with  respect  to  a;,  we  have, 
by  Art.  26,  [8], 

2a;+22/A?/  =  0. 

Solving,  D^y=-^=--. 

22/  y 

and  by  Art.  28,  [1],  the  required  equation  is 

2/-2/o= -—  (•'^-a^o)  ; 
2/0 

or,  clearing  of  fractions. 


26  DIFFERENTIAL   CALCULUS.  [ART.  30. 

but  (iCo?2/o)  is  on  the  curve,  hence 

and  we  have  ajo^  +  2/o2/  =  ^? 

the  famiUar  form  of  the  equation. 

Examples. 

(1)  Find  the  equation  of  the  normal  at  (iCo,?/o)  in  the  circle ; 
of  the  tangent  and  the  normal  at  (xQ^yo)  in  the  ellipse  and  the 
hyperbola  referred  to  their  axes  and  centre. 

(2)  Find  at  what  angle  the  curve  y^=2ax 

cuts  the  curve  ic^  —  3  axy  -\-y^  =  0. 

Am.    Cot-i\^4. 

(3)  Show  that  in  the  curve  xi  -{-y^  =  a^ 

the  length  of  that  part  of  the  tangent  intercepted  between  the 
axes  is  constant  and  equal  to  a. 


Indeterminate  Forms. 

30.  When,  under  the  conditions  of  the  problem,  the  value  of 
a  variable  quantity  is  supposed  to  increase  indefinitely.,  that  is, 
to  increase  without  limit,  so  that  the  variable  can  be  made  greater 
than  any  assigned  value,  the  variable  is  called  an  infinitely  great 
quantity  or  simply  an  infinite  quantity,  and  is  usually  represented 
b}^  the  symbol  oo.  Since  infinite  quantities  are  variables,  they 
will  usually  present  themselves  to  us  either  as  values  of  the 
independent  variable  or  as  values  of  a  function. 

31.  By  a  value  of  a  function  corresponding  to  an  infinite  value 
of  the  variable,  we  shall  mean  the  limit  approached  by  the  value 
of  the  function  as  the  value  of  the  variable  increases  indefinitely. 

Thus,  if  2/=  A 


Chap.  III.]  APPLICATIONS.  27 

and  y  approaches  the  value  a  as  its  limit  as  x  increases  indefi- 
nitel}',  the  value  of  y  corresponding  to  the  value  go  of  a;  is  a,  or 
as  we  shall  say,  for  the  sake  of  brevity, 

y  =  a  when  ic=  oo. 
Since  -  approaches  0  as  its  Umit  as  x  increases  indellnitely, 

X 

we  say  -  =  0  when  a;  =  oo, 

or,  more  briefly,  —  =  0. 

i/",  as  the  variable  increases  indefinitely  the  function  instead  of 
approaching  a  limit,  itself  increases  indefinitely,  we  shall  say 

y=  CO  when  x  =  oo, 

meaning,  of  course,  y  increases  indefinitely  when  x  increases 
indefinitely. 

32.  If,  as  the  variable  approaches  indefinitely  a  particular 
value,  the  function  increases  tvithout  limit,  we  sa\'  that  the 
function  is  infinite  for  that  particular  value  of  the  vaHable.  For 
example ;  as  the  angle  <p  approaches  the  value  90°,  its  tangent 
increases  indefinitely,  and  by  taking  (p  suflSciently  near  90°, 
tan^  can  be  made  greater  than  any  assigned  value.     So  we 

say  tan  ^  =  oo  when  ip  =  90°, 

or,  more  briefl}'  still,  tan  90°  =  oo. 

Again,  -  increases  indefinitely  as  x  approaches  zero ;  so  we 

X 

say  —  =  00  when  x=0, 

X 

or  simply  -=  oo. 

The  student  can  easily  convince  himself,  by  a  little  consideration, 


28  DIFFERENTIAL  CALCULUS.  [Art.  33. 

that  our  definition  of  infinite  is  entirely  consistent  with  the  ordi- 
nary use  of  the  term  in  algebra,  trigonometry,  and  analytic 
geometry. 

33.  The  expressions,  -,  -^,  and  0  x  oo,  are  called  indeter- 
minate forms ;  and  as  they  stand,  each  of  them  may  have  any 
value  whatever  ;  for  consider  them  in  turn  ;  —  By  the  ordinary 
definition  of  a  quotient  as  "  a  quantity  that,  multiplied  by  the 

divisor,  will  produce  the  dividend,"  -  may  be  anj^thing,  as  any 

quantity  multiplied  by  0  will  produce  0. 

So,  too,  -^  ma}^  have  any  value,  as  obviously  any  given  quan- 
tity multiplied  by  a  quantity  that  increases  without  limit  will 
give  a  quantity  increasing  without  limit. 

That  0  X  00  is  indeterminate  is  not  quite  so  obvious  ;  for,  since 
zero  multiplied  by  any  quantity  gives  0,  it  would  seem  that  zero 
multiphed  by  a  quantity  which  increases  indefinitely  must  still 
give  zero,  as  is  indeed  the  case  ;  and  it  is  only  when  0  x  oo  pre- 
sents itself  as  the  limiting  value  of  a  product  of  two  variable 
factors,  one  of  which  decreases  as  the  other  increases,  that  we 
can  regard  it  as  indeterminate.  In  this  case  the  value  of  the 
product  will  depend  upon  the  relative  decrease  and  increase  of 
the  two  factors,  and  not  merely  upon  the  fact  that  one  ap- 
proaches zero  as  the  other  increases  indefinitelj'. 

It  is  only  when  -,  -^,  and  0  x  oo  occur  in  particular  problems 

as  limiting  forms,  that  we  are  able  to  attach  definite  values  to 
them. 


34.   Each  of  the  forms -g  and  0  x  oo,  as  we  shall  soon  see, 
can  be  easily  reduced  to  the  form  -,  and  this  form  we  shall  now' 
proceed  to  stud3\ 

If  fx  =  0  and  Fx=0  when  a;  =  a, 

fx 
the  fraction  *i— ,  which  is,  of  course,  a  new  function  of  x,  assumes 
Fx 


Chap.  III.] 


APPLICATIONS. 


29 


the  indeterminate  form  -  when  ic  =  a,  and  the  limit  approached 

by  the  fraction  as  x  approaches  a  is  called  the  true  value  of  the 
fraction  when  ic  =  a,  and  can  generally  be  readily  determined. 


By  h}^othesis  /«  =  0  and  Fa  =  0, 


fx 
hence  we  can  throw  —  into  the  form 


fx-fa 


,  for  in  so  doing  we 


Fx  Fx  —  Fa 

are  subtracting  0  from  the  numerator  and  0  from  the  denomina- 
tor of  the  fraction.  Again,  we  can  divide  numerator  and  de- 
nominator by  a;  —  a  without  changing  the  value  of  the  fraction  ; 


therefore 


and  the  true  value  of 


Fx 


fx-fa 


Fx-Fa 
x  —  a 


[fif\      =  liniit  r^l     limit 


'p- 

-fa' 

X  - 

-a 

Fx- 

-Fa 

x  —  a   J 


But  cc  — a,  being  the  difference  between  two  values  of  the 
variable,  is  an  increment  of  x;  fx—fa,  being  the  difference 
between  the  values  of  fx  which  correspond  to  x  and  a,  is  the 
corresponding  increment  of  the  function,  hence 


limit 

x=a 


'fx-fa' 


=  [A^].= 


and  in  the  same  way  it  can  be  shown  that 

a;=al_   x  —  a_\     ■-    "^      jx-a^ 

fx  [D  fx'\  _ 

wherefore  the  true  value  of  ^L-  when  a?  =  a  is  Jr^^-L^,""**.     We 

Fx  \_D,Fx^,^, 

have  then  only  to  differentiate  numerator  and  denominator^  and 


30 


DIFFERENTIAL  CALCULUS. 


[Art.  34. 


substitute  in  the  new  fraction  a  for  x,  in  order  to  get  the  true 
value  required.  It  may  happen  that  the  new  fraction  is  also 
indeterminate  when  x=a;  if  so,  we  must  apply  to  it  the  same 
process  that  we  appUed  to  the  original  fraction. 

The  student  will  observe  that  this  method  is  based  upon  the 

supposition  that  /<x  =  0  and  Fa  =  0, 

so  that  it  is  only  in  this  case  that  we  have  established  the  relation 


Examples. 
Find  the  true  values  of  the  following  expressions :  — 

<■>  [i^]..,- 

(2)   ra^  +  3ar'-7a^-27a!-18" 
.3.    rxi-l  +  (x-l)f\ 

,,.  rxi-i  +  {x-i)r\ 

^>l    v(^-i)    Li 
r    i-v(i-'')  ^1  . 

Lv(i+«')-v(i+^)J.=« 


eg-,    r V(a^)  -  V(a)  +  V(a!  -  a)1 


(^>    t 


-6x^-\-Sx-3 


Ans. 


A71S,    10. 


Ans. 


Ans. 


Ans.    0. 

Ans.    1. 
1 

Ans.    00. 


35.    If  fx  and  Fx  both  increase  indefinitely  as  x  approaches 
the  value  a,  or,  as  we  say  for  the  sake  of  brevity,  if 


Chap.  III.]  APPLICATIONS.  31 

fa=  oo  and  Fa  =  oo, 

we  can  determine  the  true  value  of    ^        by  first  throwing  the 


m.j'- 


fraction  •—  into  the  equivalent  form  — ,  which  assumes  the  form 
Fx  l_ 

-  when  x=a,  and  may  be  treated  by  the  method  just  described. 

If  fx  =  0  and  Fx=  oc  when  x  =  a, 

the  true  value  of  l^fx.Fx']^^^  can  be  determined  by  throwing 

fx  0 

fx.Fx  into  the  equivalent  form  ^,  which  assumes  the  form  - 

when  x  =  a. 


Maxima  and  Minima  of  a  Continuous  Function. 

36.  A  variable  is  said  to  change  continuously  from  one  value 
to  another  when  it  changes  gradually  from  the  first  value  to  the 
second,  passing  through  all  the  intermediate  values. 

A  function  is  said  to  be  continuous  between  two  given  values 
of  the  variable,  when  it  has  a  single  finite  value  for  every  value 
of  the  variable  between  the  given  values,  and  changes  gradually 
as  the  variable  passes  from  the  first  value  to  the  second. 

37.  If  the  function  is  increasing  as  the  variable  increases^  the 
increment  Jy,  produced  by  adding  to  a;  a  positive  increment  Ax^ 

will  be  positive  ;  -^  will  therefore  be  positive,  and   li^^i*     -J- 

Ax  Ax^O  L^^J 

will  also  be  positive  ;  that  is,  D^y  will  he  positive. 

If  a  function  decreases  as  the  variable  increases^  the  increment 
Jy,  produced  by  giving  x  a  positive  increment  Ax^  will  be  nega- 
tive :  -^  will  therefore  be  negative,  and  li^it     _5(    ^^^  ^^^  '\qq 

'Ax  ^         ^         Jx=Ol_Ja;J 

negative  ;  that  is,  D^y  will  he  negative. 


32  DIFFERENTIAL  CALCULUS.  [Art.  38. 

Since  D^y^  being,  as  we  have  seen,  itself  a  function  of  a;,  may 
happen  to  be  positive  for  some  values  of  x  and  negative  for 
others,  it  would  seem  that  the  same  function  may  be  sometimes 
increasing  and  sometimes  decreasing  as  the  variable  increases, 
and  this  is  often  obviously  the  case.  For  example ;  sin  (f  in- 
creases as  (f  increases,  while  <p  is  passing  through  the  values 
between  0°  and  90° ;  but  it  decreases  as  (p  increases,  while  (p  is 
passing  through  the  values  between  90°  and  180°. 

38.  Not  only  does  an}^  particular  value  of  the  derivative  of  a 
function  show  by  its  sign  whether  the  function  is  increasing  or 
decreasing  with  the  increase  of  the  variable,  but  it  shows  by  its 
numerical  magnitude  the  rate  at  which  the  function  is  changing 
in  comparison  with  the  change  in  the  variable  as  the  latter  is 
passing  through  the  corresponding  value. 

For  example  ;  when  a;  =  2, 

D^si?  or  2x  equals  4,  and  this  shows  that  when  x  increasing  is 
passing  through  the  value  2,  its  square  is  increasing  four  times 
as  fast. 

For  if  Ax  and  Ay  are  corresponding  increments  of  the  variable 

and  the  function,  starting  from  a  particular  value  Xq  of  ic,  -^  may 

Ax 
be  regarded  as  the  mean  rate  of  change  in  y  compared  with  the 

change  in  cc,  and  J^^i*     -^    will  then  show  the  actual  rate  of 
Jic=0  \_Ax_\ 

change  at  the  instant  x  passes  through  the  value  Xq, 

39.  If,  as  the  variable  increases,  the  function  increases  up  to 
a  certain  value  and  then  decreases^  that  value  is  called  a  maxi- 
mum value  of  the  function. 

If,  as  the  variable  increases,  the  function  decreases  to  a  cer- 
tain value  and  then  increases^  that  value  is  called  a  minimum 
value  of  the  function. 

In  these  definitions  of  maximum  and  minimum  values,  the 
variable  is  supposed  to  increase  continuously. 

As  a  maximum  value  is  merely  a  value  greater  than  the  values 


Chap.  III.] 


APPLICATIONS. 


33 


immediately  before  and  immediately  after  it,  a  function  may 
have  several  different  maximum  values;  and,  for  a  like  reason, 
it  may  have  several  different  minimum  values.     If 

y=fa 

be  the  equation  of  the  curve  in  the  figure,  the  ordinates  yi  and 
2/2  are  maximum  values  of  y^  2/3  ^^^  2/4  ^^^  minimum  values  of  y. 


40.  In  the  folloiving  discussion  we  shall  suppose  throughout 
that  the  variable  continually  increases.  Then,  as  at  a  maximum 
value,  the  function  by  definition  changes  from  increasing  to 
decreasing,  its  derivative  must,  by  Art.  37,  be  changing  from  a 
positive  to  a  negative  value  ;  and  if  the  derivative  is  a  continu- 
ous function  of  the  variable  in  the  neighborhood  of  the  value  in 
question,  it  can  change  from  a  positive  to  a  negative  value  only 
by  passing  through  the  value  zero. 

Since,  at  a  minimum  value,  the  function  by  definition  changes 
from  decreasing  to  increasing,  its  derivative  must  be  chang- 
ing from  a  negative  to  a  positive  value,  and  must  therefore  be 
passing  through  the  value  zero,  provided  that  it  is  a  continuous 
function  of  the  variable  in  the  neighborhood  of  the  value  in 
question. 


41.  Confining  ourselves  for  the  present  to  the  case  ichere  the 
derivative  is  a  continuous  function^  we  can  say  then,  that  if  y  is 
a  function  of  a;,  any  value  ^of  s.  corresponding  to  a  maximum 
or  a  minimum  value  of  y  must  make  D^y  zero.  This  can  also 
be  seen  from  the  figure  of  Art.  39.  For,  at  the  points  A^  J5,  O, 
and  Z),  the  tangent  to  the  curve  is  parallel  to  the  axis  of  X,  and 
therefore  at  each  of  these  points  D^y^  which  is,  by  Art.  27,  the 


34  DIFFERENTIAL  CALCULUS.  [Art.  42. 

tangent  of  the  inclination  of  the  curve  to  the  axis,  must  equal 
zero. 

Of  course  it  does  not  follow  from  the  argument  just  presented, 
that  every  value  of  x  that  makes  D^y  =  0  must  correspond  either 
to  a  maximum  or  a  minimum  value  of  y ;  and  it  is  evident,  from 
the  figure  just  referred  to,  that,  at  the  point  E,  the  tangent  is 
parallel  to  the  axis  of  X,  and  D^y  is  zero,  although  y^  is  neither 
a  maximum  nor  a  minimum. 

42.  In  order  to  ascertain  the  precise  nature  of  the  value  of  y 
corresponding  to  a  given  value  of  x  which  makes  D^y  zero,  we 
need  to  know  the  sign  ofD^yfor  values  of  x  just  before  and  just 
after  the  value  in  question,  and  this  can  generally  be  determined 
by  noting  the  value  of  the  derivative  o/D^y,  which  we  can  always 
find,  as  D^y  itself  is  a  function  of  a;,  and  can  be  difierentiated. 

43.  D^  (D^y)  is  called  the  second  derivative  ofy  with  respect 
to  X,  and  is  denoted  by  DJ^y.  D^  (DJ^y)  is  called  the  third  de- 
rivative of  y  with  respect  to  a;,  and  is  denoted  by  D^^y ;  and  in 
general,  if  n  is  any  positive  whole  number,  D^  (Z)/-^y)  is  called 
the  nth  derivative  of  ^'  with  respect  to  a?,  and  is  denoted  by  Dj'y. 

44.  Example.  Required  the  nature  of  the  value  of  a;^  — a^ 
corresponding  to  the  value  0  of  x. 

Lei  y  =  a?  —  x^: 

D^y  =  ^a?-2x, 

DJ^y=6x-2; 

[A2/]x=o=0, 

[A^2/]_o=-2. 

Since  D^^y  is  negative  when  x  =  0,  D^y  must  have  been  de- 
creasing as  X  passed  through  the  value  zero,  and  as 


UNIVERSITY  ^ 


Chap.  III.]  APPLICATIONS.  35 

D^y  must  have  been  positive  before  a;  =  0,  and  negative  after 
x  —  0\  therefore,  y  must  have  been  increasing  before  ic  =  0,  and 
decreasing  after  ic  =  0,  and  must  consequently^  have  a  maximum 
value  when  x=0.  To  confirm  our  conclusion,  let  us  find  the 
values  of  »^  —  a^  when  a;  =  —  .1,  when  x  =  0,  and  when  a;  =  .1 : 

and  the  value  corresponding  to  a;  =  0  is  the  greatest  of  the  three. 

45.  If  [^-2/].=.„=0 

"»<1  [J5/2/].=.„>0, 

D^y  must  have  been  increasing  as  x  passed  through  the  value 
Xq  ;  and,  therefore,  since  D^y  =  0  when  x  =  Xq,  it  must  have  been 
negative  before  x  =  Xq  and  positive  after  a?  =  a^o :  y  then  must 
have  been  decreasing  before  x=:Xfi  and  increasing  after  x  =  Xq^ 
and  so  must  be  a  mirn'mnTn  when  x  =  a^o. 

46.  If  [A2/].=.„=0 

and  [A^2/]x=x„=0. 

we  must  find  the  value  of  DJ'y  before  we  can  decide  on  the  nature 
ofyo.    Suppose  [.I>,y']x=x„=*>y 

[DJylx=x=o, 

and  [i>/j']x=x„<0. 

As  \^Dx^y'\x=x  is  negative,  DJ'y  must  have  been  decreasing  as 
X  passed  through  the  value  Xq,  and  being  0  when  x  =  Xq,  must 


36 


DIFFEEENTIAL  CALCULUS. 


[Art.  47. 


have  been  positive  before  and  negative  after.     D^y  therefore 
must  have  been  increasing  before  x  =  Xq  and  decreasing  after ; 


and  as 


[A2/]^=..,=  0, 


it  must  have  been  negative  both  before  and  after  x  =  Xq.  The 
function  ?/,  then,  must  have  been  decreasing  both  before  and 
after  x  =  Xq,  and  2/0  is  neither  a  maximum  nor  a  minimum. 

Examples. 
(1)    Show  that  if        [l>.2/]^=^^  =0, 

2/0  is  neither  a  maximum  nor  a  minimum. 


(2)    If 

[•DxV].=x<,=0, 

and 

[A«3/].=.,<0, 

2/0  is  a  maximum. 

(3)    If 

and 

[^.^^]x=.„>0. 

?/o  is  a  minimum. 

47.  The  preceding  investigation  suggests  the  following  method 
of  finding  the  values  of  the  variable  corresponding  to  maximum 
or  minimum  values  of  the  function.  Differentiate  the  function 
and  find  what  values  of  x  will  make  the  first  derivative  zero. 
This  may,  of  course,  be  done  by  writing  the  derivative  equal  to 
zero,  and  solving  the  equation  thus  formed.  Substitute  for  x, 
in  turn,  in  the  second  derivative,  the  values  of  x  thus  obtained, 
and  note  the  signs  of  the  results.  Those  values  of  x  which  make 
the  second  derivative  positive  correspond  to  minimum  values  of 


Chap.  III.]  APPLICATIONS.  37 

the  function,  and  those  that  make  the  second  derivative  nega- 
tive^ to  maximum  values  of  the  function.  If  any  make  the 
second  derivative  zero,  they  must  be  substituted  for  x  in  the 
third  derivative,  and  the  result  interpreted  by  the  method  of 
Art.  46. 

Examples. 

Find  what  values  of  x  give  maximum  and  minimum  values  of 
the  following  functions  :  — 

(1)  u=2a^-21x'-\-S6x-20. 

Ans.   x=l,  max.  ;  ic=  6,  min. 

(2)  u  =  x^-9x^-^15x-3. 

Ans.  ic  =  1,  max.  ;  a;=  5,  min. 

(3)  u  =  3a^-126a^+2160x. 

Aris.   Max.  when  a;  =  —  4  or  3  ; 
min.  when  a;  =  —  3  or  4. 

(4)  Showthat       u  =  a^- 3x^-\-6x-\-7 

has  neither  a  maximum  or  a  minimum  value  ;  and  that 

w  =  ic^  —  5x*  -\-  5a^  —  1 
is  neither  a  maximum  nor  a  minimum  when  x  =  0. 

(5)  A  person  in  a  boat,  three  miles  from  the  nearest  point  of 
the  beach,  wishes  to  reach,  in  the  shortest  possible  time,  a  place 

B 3 


five  miles  from  that  point,  along  the  shore.  Supposing  he  can 
walk  five  miles  an  hour,  but  can  row  only  four  miles  an  hour, 
required  the  point  of  the  beach  he  must  pull  for. 


38  DIFFERENTIAL  CALCULUS.  [Art.  48. 

With  the  notation  in  the  figure,  the  distance  rowed  is  ^/ (oc^ -{- 9) 

miles,  the  distance  walked  is  5  —  aj  miles,  and  u,  the  whole  time 

taken,  is  evidently 

V(a;^  +  9)   ,  6-x 
u  =  . 1 R—  hours, 

and  X  must  have  a  value  that  will  make  u  a  minimum. 

X  1 


4V(a^4-9)      5' 

^  9 

4(a^+9)l* 


Solving  ^ i  =  0, 

we  get  X  =  ±  4  ; 

but,  on  substituting  these  values  of  x  in  turn  in  the  expression 
for  Dj,u,  we  see  that  a;  =4  is  the  onl}'  value  which  will  make 
D^u  =  0,  since  we  must  take  the  positive  value  of  ^(ar^-f  9), 
from  the  nature  of  the  case,  as  it  represents  a  distance  traversed. 
Remembering  this  fact,  we  find 

L       X       Jx-4  ^QQ, 

and  u  then  is  a  minimum  when  a;  =4,  and  the  landing-place 
must  be  one  mile  above  the  point  of  destination. 


48.  In  problems  concerning  maxima  and  minima,  the  func- 
tion u  can  often  be  most  convenientlj'  expressed  in  terms  of  two 
variables,  x  and  v,  which  are  themselves  connected  b}^  some 
equation,  so  that  either  may  be  regarded  as  a  function  of 
the  other.  In  this  case,  of  course,  u  can,  by  elimination,  be 
expressed  in  terms  of  either  variable,  and  treated  by  the  usual 
process.  It  is  generally  simpler,  however,  to  diflTerentiate  w, 
regarding  one  of  the  variables,  a;,  as  the  independent  variable, 
and  the  other  as  a  function  of  it,  and  then  to  substitute  for  D^y 


Chap.  III.]  APPLICATIONS.  39 

its  value  obtained  from  the  given  equation  between  ^  and  y  by 
the  process  suggested  in  Art.  29. 


Examples. 

(1)    Required  the  maximum  rectangle  of  given  perimeter. 
If  a  be  the  given  perimeter,  we  have 


y 

X 


2x-^2y  =  a;  (1) 

and  the  area  u  =  xy.  (2) 

Differentiate  (1)  with  respect  to  «,  and  we  have 

2  +  2A2/=0, 

whence  D^y=  —  1;  (3) 

D,u  =  xD,y-\-y=  -x-\-y,  by  (3), 

DJ'u^  -l-^D^y=-l-l=  -2,  by  (3). 

D^u  =  Oifx  =  y, 

and  DJ^u  is  negative  ;  therefore  the  required  maximum  rectangle 
is  a  square. 

(2)  Prove  that  of  all  circular  sectors  of  given  perimeter  the 
greatest  is  that  in  which  the  arc  is  double  the  radius. 

(3)  A  Norman  window  consists  of  a  rectangle  surmounted 
by  a  semicircle.  Given  the  perimeter,  required  the  height  and 
breadth  of  the  window  when  the  quantity  of  light  admitted  is  a 
maximum.  Aiis.   Height  and  breadth  must  be  equal. 

49.    After  finding  the  values  of  ic  which  make 

it  is  often  possible  to  discriminate  between  those  corresponding 
to  maximum  values  of  u  and  those  corresponding  to  minimum 
values  of  u  by  outside  considerations  depending  upon  the  nature 


40  DIFFERENTIAL   CALCULUS.  [Art.  50. 

of  the  problem,  and  so  to  avoid  the  labor  of  investigating  the 
second  derivative. 

Examples. 

(1)  Prove  that  when  the  portion  of  a  tangent  to  a  circle  in- 
tercepted between  a  pair  of  rectangular  axes  is  a  minimum  it  is 
equal  to  a  diameter. 

(2)  Determine  the  greatest  cylinder  of  revolution  that  can  be 
inscribed  in  a  given  cone  of  revolution. 

Ans.    K  6  be  the  altitude  of  the  cone  and  a  the  radius  of  its 

base,  the  volume  of  the  required  cylinder  =  —  iza^b. 

(3)  Determine  the  cylinder  of  greatest  convex  surface  that 

can  be  inscribed  in  the  same  cone.  Ans.    Surface  =- — . 

2 

(4)  Determine  the  cylinder  of  greatest  convex  surface  that 
can  be  inscribed  in  a  given  sphere.        Ans.   Altitude  =  r  ^(2) . 

(5)  Determine  the  greatest  cone  of  revolution  that  can  be 
inscribed  in  a  given  sphere.  Ans.   Altitude  =  -  r. 

o 

(6)  Determine  the  cone  of  revolution  of  greatest  convex  sur- 
face that  can  be  inscribed  in  a  given  sphere. 

4 
Ans.   Altitude  =  -  r. 
3 

Integration. 

50.  We  have  seen  (Art.  12)  that  when  a  body  moves  accord- 
ing to  any  law,  if  -y,  t,  and  s  are  the  velocity,  time,  and  distance 
of  the  motion  respectively,  v=DtS. 

Suppose  we  have  an  expression  for  the  velocity  of  a  body  in 
terms  of  the  time  during  which  it  has  been  moving,  and  want 
to  find  the  distance  it  has  traversed.  For  example  ;  the  velo- 
city of  a  falling  body  that  has  been  falling  t  seconds  is  always 
gt,  where  g  is  constant  at  any  given  point  of  the  earth's  surface : 
required  the  distance  fallen  in  t  seconds. 


Chap.  III.] 


APPLICATIONS. 


41 


This  distance  is  evidently  a  function  of  i,  for  a  change  in  the 
number  of  seconds  a  body  falls  changes  the  distance  fallen. 
Represent  this  function  by  s  ;  then,  as 


we  have 


V=DtSy 


that  is,  the  distance  is  that  function  of  t  which  has  gt  for  its 
derivative;  and  to  solve  the  problem  we  have  to  find  the  func- 
tion wheyi  its  derivative  is  given. 


51.  Having  given  the  equation  y=fx  of  a  curve  (rectangular 
coordinates),  required  the  area  boimded  by  the  curve,  the  axis 
ofKy  a  fixed  ordinate  yo,  and  any  second  ordinate  y. 


y 


AA 


^y 


^x     d 

This  area,  A,  is  obviously  a  function  of  ic,  the  abscissa  corre- 
sponding to  the  second  bounding  ordinate  ?/,  for  a  change  in  x 
changes  A.  Let  us  see  if  we  cannot  find  the  value  of  D^A. 
Increase  x  by  Ax,  and  represent  the  corresponding  increments 
of  A  and  y  by  A  A  and  Ay.     From  the  figure,  the  area 

acdf<i  acdb  <  ecdb  ; 

but  the  area  of  the  rectangre 

acdf=  yAx, 
the  area  of  the  rectangle 

ecdb  =  (2/  +  Ay)  Ax, 
and  acdb  =  A  A ; 

hence  yAx  <  J-4  <  (y  +  Ay)  Ax ; 


42 


DIFFERENTIAL  CALCULUS. 


[Art.  52. 


and  we  want  ^^^^  f^l.     Divide  by  Ax, 


and 


nx 


AA 


That  is,  ^^^^^  always  lies  between  y  and  y  +  Ay\  and  as  they  ap- 

Ax 

proach  the  same  limit,  2/,  as  Ja;=0,   .  ^A must  be  1/,  and 


we  have 


D,A  =  y=fx; 


and  to  solve  the  problem  completely,  we  have  to  find  a  function 
from  its  derivative. 


52.  Having  given  the  equation  y=fx  of  a  curve  (rectangular 
coordinates),  required  the  length  of  the  arc  between  a  fixed 
point  (xo,yo)  of  the  curve  and  any  second  point  (x,y) . 

This  length  is  obviously  a  function  of  the  position,  and  therefore 


■) 

A 

ZV 

^y 

^x 

M 

r^ 

^■^-^ 

i:^ 

y 

vl 

^0 

X 

Ax 

of  the  coordinates  of  the  second  point ;  and  as  the  equation  of 
the  curve  enables  us  to  express  y  in  terms  of  ic,  we  can  consider 
the  length  s  a  function  of  x.  Let  us  see  if  we  can  find  its  deriva- 
tive. Increase  x  by  Ax  and  represent  the  corresponding  incre- 
ments of  s  and  y  by  As  and  Ay  respectively.     We  see  from  the 


figure  that 


PQ<As<^PN-\-Nq, 


PN  being  the  tangent  at  P. 


PQ=V{Axy+{AyY, 


Chap.  III.] 

APPLICATIONS. 

P^=  Jx .  sec  T. 

NQ=Ay-MN. 

MN=Ax.i2,iir. 

hence 

NQ=Ay-JxtanT, 

and  we  have 

43 


V(^ic)^H-  (4v)^<  ^5<  ^^  sec T-\-Jy^Jx  tan t. 
Divide  by  Ja?,  — 

and        }^™^*     sec  T  4-  --^  —  tan  T   =  sec  r  +i)a!2/  —  tan  r. 
Ax=0\_  Ax  J 

But  we  know,  Art.  27,  [1],  that 

tan  r  =  D^y ; 

and  by  trigonometry, 

sec2  r  =  1  +  tan2  r  =  1  4-  {B^yY, 


hence 


umt^  l^sec  r  +  2  _  tan  rj  =  Vl+CAj/)^  +  ^.V  -  D.V, 


limit 

Ja? 


or  =-Jl  +  {D,yy. 

As 
As  — -  lies  always  between  two  quantities  which  have  the  same 

ijX     

limit,  Vl  +  {D^yY^  its  limit  must  be  Vl  -{-{D^yY,  and  we  have 


D^y  can  be  found  from  the  given  equation,  and  therefore 


44  DIFFERENTIAL   CALCULUS.  [Art.  55. 


Vl  +  {D^yy  can  be  determined.  We  can  then  regard  D^s 
as  given,  and  again  we  are  required  to  obtain  a  function  from 
its  derivative. 

53.  To  find  a  function  from  its  derivative  is  to  integrate,  and 
the  function  is  called  the  integral  of  the  given  derivative. 
Thus  the  integral  of  2x  is  a^  +  O,  where  C  is  any  constant,  for 
Z)^(a^4-  (7)  is  2x.  In  other  words,  if  i/  is  a  function  of  x,  that 
function  of  x  which  has  y  for  its  derivative  is  called  the  integral 
of  y  with  respect  to  x,  and  is  indicated  by  f^y,  the  symbol  f^ 
standing  for  the  words  integral  with  respect  to  x. 

54.  Since  the  derivative  of  a  constant  is  zero,  we  may  add 
any  constant  to  a  function  without  affecting  the  derivative  of  the 
function  ;  so  that  if  we  know  merely  the  value  of  the  derivative,  the 
function  is  not  wholly  determined,  but  may  contain  any  arbitrary, 
i.e.,  undetermined,  constant  term.  In  special  problems,  there 
are  usually  sufficient  additional  data  to  enable  us  to  determine 
this  constant  after  effecting  the  integration. 

55.  Since  integration  is  defined  as  the  inverse  of  differentia- 
tion, we  ought  to  be  able  to  obtain  a  partial  set  of  formulas  for 
integrating  by  reversing  the  formulas  we  have  already  obtained 
for  differentiating.     Take  the  formulas  — 

D^ax=a ; 
D,ay  =  aD,y', 
D^x"  =  Tix"-^ ; 

D,{u-\-v-^w-\-&G.)  =  D^u-\-D^v-{-D^w-\'&c.; 
and  we  get  immediately  — 
fA  =  x+C;  (1) 

f^a  =  ax-\'G;  (2) 


Chap.  III.]  APPLICATIONS.  45 

f,aD,y=:ay-{-C;  (3) 

f^nx^-^  =  x^ -\- C ;  (4) 

f;,{D^u  +D^v  4-  D^w  +  &c.)  =  u-\-v  +  io  +  &Q.  +  C;  (5) 

where  C  in  each  case  is  an  arbitrary  constant. 
The  forms  of  the  last  three  can  be  modified  with  advantage. 

In(3),caU  D^y  =  u; 

then  y  =f^u, 

and  (3)  becomes  f^au  =  af^u-\-C.  (6) 

By  the  aid  of  (6) ,  (4)  can  be  written, 

Change  n  into  n  + 1 ,  and  we  get 

{n-\-l)f,x-  =  x-  +  ^  +  C, 

or  Xaf  =  ^+C,  (7) 

71  +  1 

where  C  is  any  arbitrary  constant,  although,  strictly  speaking, 
different  from  the  (7  just  above. 

In  (5) ,  let  D^u  =  ?/,  D^v  =  z,  &c., 

then  u=f^yj  v=f^z,  &c. 

and  A(y  +  z+&c.)=f,y+f,z-j-&c,-hC; 

or,  tJie  integral  of  a  sum  of  terms  is  the  sum  of  the  integrals  of 
the  terms. 

56.    We  can  now  solve  the  problem  stated  in  Art.  50.     The 
velocity  of  a  falling  body  at  the  end  of  t  seconds  is  gt  feet,  g 


46  DIFFERENTIAL  CALCULUS.  [Art.  67. 

being  a  constant  number ;  required  the  distance  fallen  in  t  sec- 
onds. We  have  seen  that,  if  v,  t^  and  s  are  the  velocity,  time, 
and  distance  respectively,     v=DtS; 

hence  s=ftV. 

Here  s=ftgt  +  G; 

but  by  Art.  55,  (6)  and  (7), 

and  in  this  case  we  can  readily  determine  C,  for  when  the  body 
has  been  falling  no  time,  it  has  fallen  no  distance,  :o  s  must 
equal  zero  when  ^  =  0,  and  we  have 

0  =  i^(0)2  +  O=04-C,  and  (7=0; 

and  our  required  result  is     s  =  ^gf. 

57.  Required  the  area  intercepted  by  the  curve  2/^=4jc,  the 
axis  of  X,  and  the  ordinate  through  the  focus. 

From  the  form  of  the  equation  we  know  that  the  curve  is  a 
parabola  with  its  vertex  at  the  origin  and  its  focus  at  the  point 
(1,0) .  The  initial  ordinate  in  this  case  is  evidently  the  tangent 
at  the  vertex. 

If  A  is  the  required  area,  DgA  =  y,  (Art.  51) , 

then  A=f^y. 

y=  2Vi  =  2iC5; 

hence         A  =f^2xk  =  2f,xh  =  M  +(7=  -  a;l  +  (7. 

A  stands  for  the  area  terminated  by  the  ordinate  correspond- 
ing to  any  abscissa  x. 

It  is  obvious  from  the  figure  that  if  we  make  a;  =  0,  the  ter- 
minating ordinate  y  will  coincide  with  the  initial  ordinate  through 
the  origin,  and  A  will  equal  zero.   So  we  can  readily  determine  C, 


Chap.  III.] 


APPLICATIONS. 


47 


for  we  have 
so  that 

and 


y 

y 

0 

.      P         X                  X 

K 

^  =  0  if  a;  =  0  ; 
0  =  |01-hO  =  C, 


3 


If  ic  =  1,  as  it  must  in  order  that  y  may  pass  through  the  focus, 

^  =  -,  the  required  area, 

o 


Examples. 

(1)  Find  the  area  bounded  by  the  curve  x^=4y,  the  axis  of 
X,  and  the  ordinates  corresponding  to  the  abscissas  2  and  8. 

Arts.   42. 

(2)  Prove  that  the  area  cut  off  from  a  parabola  by  a  double 
ordinate  is  two-thirds  of  the  circumscribing  rectangle. 

(3)  Required  the  area  intercepted  between  the  curves  y^=  4  ax 


and  a^=  4  ay. 


Arts. 


16  a^ 


(4)  Find  a  formula  for  the  area  bounded  by  a  curv^e  x=fy^ 
the  axis  of  F,  and  two  lines  parallel  to  the  axis  of  abscissas. 

Ans.   A=fyX-\-C. 

(5)  Find  a  formula  for  the  area  intercepted  by  a  curve  y  =fx, 
the  axis  of  X,  and  two  ordinates  (oblique  coordinates) . 

Ans.   A  =  sin  lof^y  -{-C,  oj  being  the  inclination  of  the  axes. 


48  DIFFERENTIAL   CALCULUS.  [Art.  58. 

(6)    Prove  that  the  segment  of  a  parabola  cut  off  by  any  chord 
is  two-thirds  of  the  circumscribing  parallelogram. 

58.   Required  the  length  of  the  portion  of  the  line 

4,x-Sy-\-2  =  0  (1) 

between  the  points  having  the  abscissas  1  and  4. 


We  have  seen  that  D^s  =  VT+C^^^  Art.  52, 

where  s  is  the  length  of  an  arc ; 
hence  s=f,^/lT{R^\ 

From  (1)  we  get  4-3  D^y  =  0, 


Vi-f(i>.2/)'  =  l; 

and  therefore  s=Jl^  =  ^x-\-C,         where  s  stands  for 

the  length  of  the  arc  from  the  first  point  to  any  second  point  whose 
abscissa  is  x.  If  we  make  a;  =  1 ,  the  two  points  will  coincide  and 
s  must  equal  0  ;  then  0  =  f  +  (7, 

and  s  =  f(x  — 1). 

To  get  the  required  distance,  x  must  equal  4  and  we  get  s  =  5. 

Example. 

Find  the  length  of  the  portion  of  the  line  Ax -\-  By -\- C  =  0 
between  the  points  whose  abscissas  are  Xq  and  x^. 

Ans.    V(i^(,,_^). 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS.  49 


CHAPTER  IV. 

TRANSCENDENTAL  FUNCTIONS. 

59.    In  order  to  complete  our  list  of  formulas  for  differentiat- 
ing, we  must  consider  the  transcendental  forms,  logic,  a'',  sina;,  &c. 
Let  us  differentiate.Xogx. 
By  our  fundamental  method,  we  have 

V^ogx-^^^^\^ -^ J' 

\og{x+Ax)-\ogx  ^  J_  1     r^±^1  ^  _!_  1     [i  ^  ^1  ^ 
Ax  Ax        \_     X     ]      Ax        \_         X  y 

z>.iog.=j'-:*,[j-iog(i+f)]. 

But  as  Ax  approaches  zero,  logfl  +— )  approaches  logl,  i.e., 

zero,  and  —  increases  indefinitely  ;  so  that  it  is  by  no  means  easy 
Ax 

1         /        Ax\ 
to  discover  the  limit  of  the  product  —  log  (  l  -| ] . 

LiX  \  *^  / 

This  product  can  be  thrown  into  a  simpler  form  by  introduc- 

X 

ing  m  =  —  in  place  of  Ax. 

Ax 

1         f    ,  ^A  m         /        1  \        1        /        1 V 

-7-  log   li then  becomes  —  log   H ,  or  -  logl  1 H )  • 

Ax     ^\        xj  X      ^\       mj^       X     ^\       mj 

X 

As  Ax  approaches  zero,  —  or  m  increases  indefinitely,  and 
m=oo|_ic     '^y        mJ  J 


50  DIFFERENTIAL  CALCULUS.  [Art.  60. 

and  the  value  we  have  to  investigate  is  the  value  approached  by 

(1  -j )    as  its  limit  as  m  increases  indefinitely,  which  we  in- 

mj 


,.     ,    ,       limit 
dicate  by  ^^^ 


60.   Let  us  first  suppose  that  m  in  its  increase  continues  always 
a  positive  integer.     Then  we  can  expand  ( i  -j )    by  the  Bi- 

v     my 


nomial  Theorem. 


/       IV        .  ^/l  \  .  m(m-l)/l  Y 


m(m-l)(m-2)/l  V 
1.2.3  \m) 


-\-  &c.  to  m  + 1  terms 

^1,1,  m       V___mA__W 

1         1.2  1.2.3 


1 \ A       2 \/        3 


V        mj\        mj\        m, 
+-^ ^  +  ....tom+l  terms. 

Now,  as  m  increases  indefinitely,  each  of  the  first  n  terms  of 
the  series,  n  being  any  fixed  number,  approaches  as  its  limit  the 
corresponding  term  of  the  series 


1       1.2      1.2.3      1.2.3.4  ' 

so  that  we  have  reason  to  suppose  that  there  is  some  simple  rela- 
tion between  this  latter  series  and  our  required  limit. 

61 .  To  investigate  this  question  we  shall  divide  the  first  series 
into  two  parts.  The  first  part,  consisting  of  the  first  n-\-l  terms, 
where  n  is  any  fixed  whole  number  less  than  m,  we  shall  repre- 
sent by  S ;  the  second  part,  consisting  of  the  remaining  m  —  n 
terms,  we  shall  call  E. 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS. 

Then 


61 


JS 


1,1,  m       V^7iA__W 

1         1.2  1.2.3 

\       my\       771 J       \         m  J 


/ 


^  1.2.3....n 

As  n  is  a  fixed  number,  we  have 

,=00  L    J       ^1^1.2^1.2.3^        ^1.2.3 


hmit 
m: 


■ti : — -— • —  • — — ■ — -~~ 


1.2.3 n 


n  +  \  (7i  +  l)(n+2) 


+  ....-f 


\       mj\         ^'^   /V         m   J        \  m 

(n  +  l)(w+2)(?i+3) m 


Since  n  is  less  than  7?i,  each  numerator  in  the  value  of  R  is  posi- 
tive and  less  than  1 ,  and 


^< 


1.2.3 n 


1 


71  +  1      (n-{-\Y      (71  +  1)' 
The  sum  of  the  decreasing  geometrical  series 

1 


....+ 


^         1 

(71 +1) '"-"J 


is  b}^  algebra  less  than  -  ; 
n 


71-1-1  (7^+1)2^   (71+1)^ 


1 


+ 


therefore 


R< 


71(1.2.3. ...nV 


52                             DIFFERENTIAL  CALCULUS.  [Art.  62. 

and  ^^^  [i2]< 1 -; 


and  we  have  at  last. 


imit  (^_ulY=  li^^it  p^-j      limit  ™ 


limit 


=  i+UJ,+^„+,JL,+....  +  '     1 


1      1.2    '   1.2.3      1.2.3.4  1.2.3 n 

+  something  less  than , 

w(1.2.3 n) 

n  being  any  positive  whole  number. 

Thus  we  obtain  the  relation  that  the  difference  between  our 
required  value  and  the  sum  of  the  first  n-\-l  terms  of  the  series 

^1      1.2^1.2.3^ 
is  less  than 


%(1.2.3 n) 

The  greater  the  value  of  n  the  less  the  value  of '- —  ; 

^  71.1. 2.3. ...n 

and  by  taking  a  value  of  n  sufficiently  great,  we  may  make  this 
difference  as  small  as  we  please. 

Consequently,  by  Art.  7,  our  required  value  is  the  limit  ap- 
proached by  the  sum  of  the  first  n  terms  of  the  series 


1      1.2       1.2.3  ' 

as  n  is  indefinitely  increased,  or  what  is  ordinarily  called  the  sum 
of  this  series. 

62.   The  series  1  +  -  +  -—  +  — —  + . . . .  plays  a  very  impor- 

tant  part  in  the  theor}^  of  logarithms.  It  is  generall}^  represented 
by  the  letter  e,  and  is  taken  as  the  base  of  the  natural  system 
of  logarithms.  Its  numerical  value  can  be  readily  computed  to 
any  required  number  of  decimal  places,  since  each  term  of  the 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS.  53 

series  may  be  obtained  by  dividing  the  preceding  one  by  the 
number  of  the  term  minus  one.  Carrying  the  approximation 
to  six  decimal  places,  we  have 

1. 
1. 

0.5 

0.1666666 

0.0416666  The  error  in  the  approximation  is 

0.0083333        less  than  one-eleventh    of    the    last 

0.0013888        term  we  have  used,   and    therefore 

0.0001984        cannot  affect  our  sixth  decimal  place. 

0.0000248 

0.0000027 

0.0000002 

0.0000000 


e=2.718281+,  correct  to  six  decimal  places. 

63.  Let  us  now  remove  from  m  the  restriction  we  placed  upon 
it  when  we  supposed  it  to  have  none  but  positive  integral  values, 
and  suppose  it  to  increase  passing  through  all  positive  values. 
Let  //  represent  at  an}'  instant  the  integer  next  below  m,  then 
/jt  +1  will  be  the  integer  next  above  m,  and  as  m  increases  it  will 
always  be  between  p.  and  /^.  -f- 1  ?  unless  it  happens  to  coincide  with 
/x  +  1,  as  it  sometimes  will.     We  have,  then,  in  general, 

M<m</i  +  l. 

limit  Cj+iy-, 

(JL^CC    \  fxj 

/  .    ly   v^  /^+i;        ,  limit vVfV__e_ 

[H n     = ; J  and        ^ z =  -  =  e. 


M+1  At+1 


54  DIFFERENTIAL   CALCULUS.  [Art.  64. 

(-3"=(-9'(-3' 

hence  '™'*  ('l+i-)"=e. 

m=co\       my 

Again :  let  m  be  negative,  and  represent  it  by  —  r, 

-(-,4i)-(-^). 

and  «»"  fl  +  iy 

m=oo  \        my 

-,J',ti.(-,4T)-(.^,4T)  =  -,  =  .. 

We  see,  then,  that  always 
64.   In  Art.  59  we  found  that 
We  have ,  then,  D^  log  a;  =  1  log  e . 


If  by  logo;  we  mean,  as  we  shall  always  mean  hereafter,  natural 
logarithm  of  a;,  loge  will  equal  1,  and 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS.  65 

DJogX  =  -.  [1] 


If  y=fx,  Alog2/  =  ^. 


Exponential  Functions. 
65.  Required  D^a'',  a  being  any  constant. 
Let  u  =  a' 

and  take  the  log  of  each  member, 

logw  =  a;loga. 
Take  D^  of  both  members, 


D.u 
u    =^^g^-' 

X)^w  =  'wloga, 

D^  a'' =  a*  log  a. 

[1] 

If 

a  =  e;  since  loge  =  1, 

we  have 

Ae*  =  e^ 

[2] 

Of  course, 

Z)^a^==aMogaZ)^2/, 

and 

B^e'^e^B^y, 

Examples. 
Find  B^u  in  each  of  the  following  cases  :  — 
(1)    1^  =  6=^(1  _a^).  Ans.   B,n  =  e{\-Zo^-y?^ 


56  DIFFERENTIAL  CALCULUS.  [Art.  66 

(3)    u  =  log(e^  +  e-^).  Ans.   D^u=^^^ 


.,.  X  A         n         e''(l  —  x)  —  l 

(4)    |^  =  _± — .  Ans.   D^u=     ^         ^ 


(6)    w  =  log(logic).  ^ns.    D^u  = 


xlogx 
1  1 


(7)  w  =  loff T-.  Ans.   D-u== 

(8)  u  =  af.  Ans.   D^u=x''(\ogx-}-l) 

Suggestion.     Take  the  log  of  each  member  before  differen 
tiating. 

(9)  u  =  xk.  Ans.   D^u  =  ^2(lll^ 


x" 

(10)  u  =  e^.  Ans.    D^u  =  e^e'' 

(11)  u  =  e^.  Ans.   D^u  =  e'>^x^{l-{- log x) 

(12)  u  =  x^.  Ans.    D^u  =  x^e^-^ — 

Trigonometric  Ftmctions. 

66.  In  higher  mathematics  an  angle  is  represented  numerically^ 
not  by  the  number  of  degrees  it  contains  but  by  the  ratio  of  the 
length  of  its  arc  to  the  length  of  the  radius  with  which  the  arc  is 
described. 

Thus  the  angle  0  is  said  to  be  equal  to  .?£^.     If  the  arc  is 

r 

^^     described  with  a  radius  equal  to  the  linear  unit, 
^   this  ratio  reduces  to  the  lens^th  of  the  arc.    This 


method  of  measuring  an  angle  is  called  the  ciV- 
cular  or  analytic  S3^stem,  as  distinguished  from 
the  ordinary  degree  or  gradual  system. 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS.  57 

27rr 

The  value  of  360°  in  circular  measure  is  obviously or  2iv, 

r 

and  of  1°  is  — —  or  — ^.     Hence,  to  reduce  from  gradual  to  cir- 

360       180 
cular  measure^  it  is  only  necessary  to  multiply  the  given  number 

of  degrees  by  ^. 

The  circular  unit  is  evidently  the  angle  which  has  its  arc  equal 
to  the  radius,  and  its  value  in  degrees  is  easily  found.  Let  x 
represent  the  required  value  in  degrees  ;  then 

x°          r         .         180° 
and  X  = . 


360°      2;rr 

Hence,  to  reduce  from  circidar  to  gradual  measure,  we  have  only 

180 
to  multiply  the  circidar  value  by 


6  7 .    Required  D^,  sin  x . 

By  our  usual  method,  we  have 

2>.8ina^=  limit  pi^(^  +  ^a^)- sina;-| 
Ax=0\_  Ax  J' 

sin(a;  +  Jic)  —  sinic  _  sin  ic  cos  Ax  -\-  cos  a;  sin  Ax  —  sin  a; 
Ax  Ax 

GOQX  sin  Ax  —  sinic(l  —  cos  Ax) 
^  Ax 

n    '           limit  f         sin  Ax        .      1  —  cos  Ax\ 
D^smx=  ""^^^  I  cosa; — -, sma; -, 


=  cos  a; 

Ax 


Ja;=0 1_  Ax  Ax 

limit  fsinJa^n^.^^  limit  ["l-cosJx"! 
=  0\_    Ax   ]  Ja;=OL       Ax       J" 


But  as  Ax  =  0,  sin  Jx  =  0  and  cos  Ja;  =  1,  so  both  of  our  limits, 
in  their  present  form,  are  indeterminate,  and  require  special 
investigation. 


58 


DIFFERENTIAL  CALCULUS. 


[Art.  68. 


68.  Suppose  an  arc  described  from  the  vertex  of  the  angle  Jo;, 
w  K  with  a  radius  equal  to  unity,  then  this  arc 
measures  the  angle,  and  is  equal  to  Ax^  and 
the  lengths  of  the  lines,  marked  s  and  c  in 
the  figure,  are  the  sinJa;  and  cosJa;,  respec- 
tively. 


We  wish  to  find 


limit  [_£_]  and  limit  fel 
da;=0|_Ja;J         Ja;=0|_  Ax  J 


Ax=0\_Ax^         Ax-- 
QxcAx<CAB-\-BD 
by  geometry  {vide  "  Chauvenet's  Geometry,"  Book  V.  Prop,  xii.) . 
We  have  then  AD <,Ax<AB-\-BD; 

or,  since  s^AD,  and  AB+BD  =  s-\-l  —  c, 

s<iAx<Cs+l—c. 
But  s2  +  c2=l, 


and 


hence 


or 


l-c  = 


rT"c' 


1  +  c 

s<Ax<:'(l±^)±l; 
1  +  c 


s{l  +  c) 
s"  Ax^  s{l-\-c)-\- 


^>4r> 


l>-f  > 


1  +  c 


Ax      1+c  +  s' 

and  jiinit  ["  «  1  j^^gt  be  between  1  and   ^'^^^  f-I+^l  •  but 

Ax~0  [_Ax]  Ax=0  [_l-f. c  +  sj 

since,  as  Ja?  ==  0,  s  =  0,  and  c  ==  1, 


Chap.  IV.]  TRANSCENDENTAL  FUNCTIONS. 

Umit  I      i+c     I      2 


Ax 


m 

therefore 


nit  ["    1+c    1- 
=  0LH-c  +  sJ 

nit  fsinJ^I 
=0[_   Ja;   J 


limit 

Ax 


2 
=  1. 


1; 


In  like  manner,  >— —  >     ...  .    ;  .    / 

s  Ax        s(l  +  c)  +  s^ 

or  —rr-. — r>^ — > 


s(H-c)        Ax       s(l4-c)+s2' 


'     >1^> 


59 


l  +  c       Ja;       1  +  c  +  s 

rizi^l  lies   between    ^i^i^     [-^T   ^   ^^    and 
[_  Ja;  J  Ja;=0    L^  +  d 

imit  r f ],orO; 

a;=0Ll+c4-sJ 


limit 
Ja;=0 

limit 
Ja; 


therefore 


limit  [1- cos  Ja;"]_Q 
Ja;=oL       ^a;      J 


69.    Substituting  these  values  in  Art.  67,  we  have 
2)j,sina;=  cos  a;. 


(1)  Prove 

(2)  Prove 


Examples. 
Z)^cosa;=  —  sin  a;. 
Z>,tana;=  sec^a;, 
i)^ctna;=  —  csc^a;, 
Dj  sec  X  =  tan  x  sec  a;, 
D,  CSC  X  =  —  ctn  X  CSC  a;. 


from  the  relations 


,  sma; 

tana;= , 

cos  a; 


60 


DIFFERENTIAL   CALCULUS. 

ctnx  = 

1 
tanx 

seca;  = 

1 

cos  a; 

CSCiC  = 

1 

sin  a; 

vers  a;  =  1 

—  cos  a;, 

A  vers  a? 

=  sin  a;. 

[Art.  70. 


(3)    Given 
prove 

4)    Prove  D^  log  sin  a;  =  ctn  a; ; 

Z>a.logcosa;  =  — tana;; 
Djj  log  tan  a;  =  sec  a;  esc  a; ; 
Z)jjlogctna;=  —  seca^csca;; 
D^\og sec X  =  tan x  ; 
D^logcscx=  —  etna;. 

Anti-  Trigonometric  Functions. 

70.  In  trigonometry,  the  angle  lohich  has  a  sine  equal  to  x  is 
called  the  inverse  sine  or  the  anti-sine  ofx,  and  is  denoted  b}"  the 
symbol  sin~^  Hence  sin~^x  means  the  angle  which  has  x  for 
its  sine,  and  is  to  be  read  anti-sine  of  a;. 

In  the  same  way  we  speak  of  anti-cosine,  anti-tangent,  &c. 

71.  To  differentiate  sin~^a;. 

Let      '  y  =  sin~^a; ;    then  x  =  sin 2/. 

Differentiate  both  members  with  respect  to  x. 
l=cosyD^y; 


Chap.  IV.]  TBANSCEKDENTAL  FUNCTIONS.  6l 

cosy 
It  remains  to  express  cosy  in  terms  of  a;. 

sin  2/  =  flJ, 
cos^y  =  l—oc^, 

cos2/=v(i-^) ; 


nee 

^'«'"-'^=V(l-x^)- 

Examples. 

(1)   Prove 

n  on"~^T — 

/y.cos     X-      ^^j_^)- 

(2) 

D  +in~l'y  — 

/>.tan    •^-^_^^- 

(3) 

7)  pfn— I'y  — 

1+a^ 

W 

n  opp— i/y. —          ■*- 

Uj.  sec    X  —                      • 

(5) 

n  pep— i/v.  —                 ^ 

xyj.  CSC     a;  —                            ■ 

(6) 

Avers-^a;  = ?^ -. 

72.  The  anti-,  or  inverse,  notation  is  not  confined  to  trigono- 
metric functions.  The  number  which  has  x  for  its  logarithm  is 
called  the  anti-logarithm  of  x,  and  is  denoted  by  log~^x;  and, 
in  general,  if  x  is  any  function  of  y,  y  may  be  called  the  corre- 
sponding anti-function  of  x,  and  the  relation  of  y  to  a;  will  be 
indicated  by  the  same  functional  sj^mbol  as  that  which  expresses 


62  DIFFERENTIAL  CALCULUS.  [Art.  73. 

the  dependence  of  x  upon  ?/,  except  that  it  will  be  affected  with 
a  negative  exponent,  which,  however,  must  not  be  confounded 
with  a  negative  exponent  in  the  algebraic  sense.  Thus,  if  x=fy, 
we  may  write  y=f-^x. 

Any  anti-function  can  be  readily  differentiated,  if  the  direct 
function  can  be  differentiated,  and  b}^  the  method  we  have  em- 
ployed in  the  case  of  the  anti-trigonometric  functions  above. 

Let  yz=f-^x, 

then  x=fy; 

differentiate,  and  1=  Dj,fy .  D^y. 

1 


A2/  = 


A/y 


or 


DJ-'x  =  J-, 
Dyfy 


and  it  is  only  necessary  to  replace  y  in  this  result  by  its  value 
in  terms  of  x. 

73.    Since,  in  the  formula  above, 

fy  =  a;, 

we  have  D^y  —  ——  ; 

DyX 

a  result  so  important  that  it  is  worth  while  to  establish  it  by 
more  elementar}-  considerations. 

Suppose  X  and  y  connected  b}'  any  relation,  so  that  either  may 
be  regarded  as  a  function  of  the  other.  Let  Ax  and  Ay  be  cor- 
responding increments  of  x  and  y.  Then  Ax  may  be  regarded 
as  having  produced  Ay^  or  as  having  been  produced  by  J?/,  ac- 
cording as  we  regard  x  or  y  as  the  independent  variable ;  and 
on  either  hypothesis  they  will  approach  zero  together. 


By  definition,  D,y=  ^i"^^*  f— T 


Chap.  IV.] 
and 


TRANSCENDENTAL  FUNCTIONS. 


7)^=  limit  r^i. 


63 


and 


limit  [^y]_ 


^1 

Jxj 


Ay 

1 

Ax 

~  Ax' 

Ay 

1 

Ax=0\_^x_^       limit  I  ^^1       limit 


mit  r^~|       limit  fil^l 
:=0  \_Ay]      Ay=0  [jyj 


Ja;=0  I  Av  I      J?/=0 
since  Ax  and  Jy  approach  0  together. 


Therefore 


DyX 


Examples. 

Find  D^u  in  the  following  cases  :  — 


(1)  w  =  sin^a;. 

(2)  I*  =  cosmic. 

(3)  ^4  =  a;e^°^^ 

(4)  it  =  cos(sina?). 

(5)  16=  sin  (log  a;). 

(6)  u=^^^^-is.nx  +  x. 


(7)    u  =  {a'  +  x')ifm-^±. 
a 


(8)    w  =  a;sin~*£C. 


(9)    t*  =  sm^     ^ 


^Tis.   Z>a.  w  =  2  sin  a;  cos  x. 
Arts.   D^ u=  —m sin mx, 

Ans.    D^  u  =  e*=°«^(  1  —  x  sin  x) . 
Ans.   D^u=  —  cos  a;  sin  (sin  x) . 

Ans.   D^u  —  -  cos  (logo;) . 

X 

Ans.    D^u  =  tan^x 


Ans.   D^u=2xiaxr^  -^a. 
a 


V(2) 


Ans.   D^u  =  sin~ ' a;  -f- 
Ans.   D^  u 


V(i-^) 
1 


^{\-2x-x^) 


'^\  B  R  A  /^p• 

OF   THK 

T  r "NT  T-X7"P -n  CITT^ V- 


64  DIFFERENTIAL  CALCULUS.  [Art.  73 

1 


(11)    u  =  seG~^ —    /^         »  Ans.   D^u=         ^ 


(12)  w  =  sin-^V(sina;).  -d?is.   Z)^w=  ^  V(l  +  csca;) 

(13)  ..  =  tan-^^.  Ans.    D^u  =  ^ 

2 


(U)    „  =  tan-'    f-^1^55f'\ 


Chap.  V.J  INTEGRATION.  66 


CHAPTER  V. 

INTEGRATION. 

74.  We  are  now  able  to  extend  materiall}'  our  list  of  formulas 
for  direct  integration  (Art.  55) ,  one  of  which  may  be  obtained 
from  each  of  the  derivative  formulas  in  our  last  chapter.  The 
following  set  contains  the  most  important  of  these  :  — 

D^o^x  =  -  gives  f^-  =  logic. 

X  X 

D^a''  =  anoga  ''  /,a*loga  =  a*. 

D^e'  =  e'  ''  f^e'=e'. 

D^^mx  —  Qo^x  "  Xcosa;=  sina;. 

Z>jBCOSic=  —  sina;  ''  f^{—sinx)  =  co^x, 

i>a,logsina;  =  ctnic  "  Xctna;  =  logsina;. 

D^log  cos  x=  —  tan  x  "  X  ( —  tan  x)  =  log  cos  x. 

A vers-^a;  =  — — i —      "     /x    ,.^  ^       ..  =  vers-^a;. 

y(2a;  — ar)  ^{2x  —  ar) 

The  second,  fifth,  and  seventh  in  the  second  group  can  be 
written  in  the  more  convenient  forms. 


6G  DIFFERENTIAL  CALCULUS.  [Art.  76. 

r    X         a' 
loga 

Xsma;=  —  cosic; 

Xtanx=  —log  cos  a?. 

75.  When  the  expression  to  be  integrated  does  not  come  under 
any  of  the  forms  in  the  preceding  list,  it  can  often  be  prepared 
for  integration  by  a  suitable  change  of  variable^  the  new  variable, 
of  course,  being  a  function  of  the  old.  This  method  is  called 
integration  by  substitution^  and  is  based  upon  a  formula  easily 

deduced  from  D^{Fy)  =DyFy.  D^y  ; 

which  gives  immediately 

Fy=f^{D,Fy.D^), 


Let 

u^D^Fy, 

then 

Fy=fyu,    - 

and  we  have 

fyU=f{uD^y)  ; 

or,  interchanging  x  and  ?/, 

fu=f^{uD„x).  [1] 

For  example,  required       X(a  +  6a;)". 


by  [1] ; 


Let 

z  =  a-i-  bx, 

and  then 

f{a  +  bxr=/^z-=f{z^\D,x), 

but 

b      6' 

0 

hence 

rJa4-bxY='^  Lz^=^   ^^^\ 

Chap.  V.]  INTEGRATION.  67 

Substituting  for  z  its  value,  we  have 

h       ?i  + 1 

Example. 

Find/,—?—.  Ans.    llog(a4-to). 

76.  Iffx  represents  a  function  that  can  be  integrated, /(a -f- 6a;) 
can  alwa3's  be  integrated  ;  for,  if 

z=  a-\-  hx, 

then  D,x  =  - 

b 

and  fj(a  -f  bx)  =Uz  =fJzD,x  =  \fjz, 

0 

Examples. 
Find 

(1)  Xsinax.  Ans.    _1  cos  aa?. 

a 

(2)  f.cosax.  Ans.   -sinax. 

a 

(3)  f^tanax. 

(4)  y^ctnax. 

77.  Required/^ ^ . 

/ 1 =  lr 


i}-m 


Let  2==^, 

a 

then  X  =  az, 

D^x  —  a, 


68 


DIFFERENTIAL  CALCULUS. 


-  f 

a     /r     fxy 


=1/. 


=1/, 


a-"V(l-^)      «      V(l-«') 


[Art.  78. 
D.x 


=/. 


V(i-^) 

Examples. 


=  sin~^2;  =  sin~^-. 


Find 


(W   r 

1         1^ 
^ns.    -tan-^- 

^''  -^V+o.-^ 

/'o^     /•             1 

A                         1^ 

^'^  ^'^{2ax-x^) 

^ws.    vers  ^- 

78.    i^e,w..X^^j_^^,^. 

Let                                  0  =  a;  +  V(^-'  +  «'); 

then                                z  —  x  =  ^{oi^-\-a^), 

z^-2zx-^x'  =  x^  +  a\ 

2zx  =  z'-a^, 

"'-     2z    ' 

'Jix'-^a^^-z     x-z     ^'-«'- 

_z^^a'-     ^ 

2z 


2z 


D,x 


z2±a^ 
2z^ 


L 


/•       ^Z  /•       Zi  Z        T\ 

L-n—.=fz-^ ■-  D,x 


VCr^  +  a')      "V  +  a^     *'V  +  a^ 


■L 


2z     z'^-\-o?      .1 


;22  4-a2     2^2        •^--  =  log2;  =  log(a;  +  Va^  +  a^ . 


Example. 


Find/, 


V(^-a^) 


^ns.   log(a;  +  Var^  — a^). 


Chap.  V.]  INTEGRATION.  69 

79.    When  the  expression  to  be  integrated  can  be  factored^  the 
required  integral  can  often  be  obtained  by  the  use  of  a  formula 

deduced  from  D^(uv)  =  ^iD^v  -\-  vD^u, 

which  gives  uv=f^  uD^  '^  +  /x  "^Dx  ^ 

or  f^uD^v  =  uv—f^vD^u.  [1] 

This  method  is  called  integrating  by  parts. 

(a)    For  example,  required  f^logx. 

logx  can  be  regarded  as  the  product  of  logo;  by  1. 

Call  logo;  =  u  and  1  =  D^v^ 

then  D^u  =  -, 

X 

v  =  x; 
and  we  have 

fjogx  =X  1  logx  =f^uD^v  =  uv  —f^vD^u 


=  icloga;  —fx-  =  a;  log  a;  —  x. 

X 


Example. 
Find  Xa;  log  07. 
Suggestion :  Let    logo;  =  u  and  x  =  D^v. 

Ans.    -x^llosx \ 

2     \    ^        2J 

80.    Required  f^sin^x. 
Let  u  =  sin  a;  and  D^v  =  sin  a;, 

then  Z>3,?/  =  cosa;, 

v=  —  cos  a?, 
yi^sin^a;  =  —  sin  a;  cos  a;  -\-f^co3^x ; 


70  DIFFERENTIAL  CALCULUS.  [Art.  8L 

but  cos^o;  =  1  —  sin^a;, 

so  f^QO^^x=f^\—f^siv?x  =  x—f^s>\v?x 

and  y^sin^a;  =  x  —  sin  x  cos  a?  — TLsin^a;. 

2fxSiv^x  =  x  —  since  cos «. 
/^sin^ic  z=z  ^(x  —  sin  a;  cos  a;). 

Examples. 

(1)  FindXcos^a;.  Ans,   -(ic  +  sina;cosa;). 

(2)  Xsinajcosa;.  Ans.    ^^. 

81.    Very  often  both  methods  described  above  are  required  in 
the  same  integration. 
(a)    Required f^^irr'^x. 


Let 

sin~'a;  =  y, 

then 

x=  sin  2/; 

DyX  =  cos2^, 

f,sin-^x=f,y=f^7jcosy. 

Let 

u  =  y  and  D^v  =  cosy  ; 

then 

D^u  =  l, 

v=sin?/, 

and 
y^  2/ cos  2/ = 2/ sin  2/ — y^  sin  ?/ = ?/ sin  2/ + cos  ?/ = a;  sin"  *  a;  +  ^  ( 1 — a^) . 

Any  inverse  or  anti-function  can  be  integrated  by  this  method 
if  the  direct  function  is  integrable. 

(6)    Thus,  fJ-'x=f^y=f,yDJy  =  yfy^fJy 

where  y  =/"*». 


Chap.  V.J  INTEGKATION.  71 

Examples. 

(1)  FindXcos-^aj.  -4ns.   ajcos-^a?  — V(l  ~^' 

(2)  f^tsiir^x.  A71S.    xtan~^x  —  -log{l-\-x^), 

(3)  Xvers-^ic.  Ans,    (x  — 1)  vers-^a;  + VC^^— ^^)* 

82.    Sometimes  an  algebraic  transformation^  either  alone  or  in 
combination  with  the  preceding  methods^  is  useful, 

1 


(a)    Required f^- 


x^—a^ 


—  a^      2a\x  —  a     x-{-  a  J 


and,  by  Art.  75  (Ex.), 

X-J_  =  J-  [log(a; -a)- \og(x  +  a)]  =  —  log^^^. 

(b)    Bequiredf^\(^^t^, 

y^\i-xj   v(i-^)    v(i-^)    v(i-«^)' 


f^ can  be  readily  obtained  by  substituting  ?/  =  (1  —  ar*), 

and  is  —^J{l  —  a^)  ; 

nence  ^'\/(rrf)  =  ^^^"'"^  -  V(l  -  ^)  • 

(c)    Required /i- V  (<^^  ~  ^)  • 


72  DIFFERENTIAL  CALCULUS.  [Art.  83. 


a^ 


whence       A^{a?-^)  =  aHm-'^-f^        f  by  Art.  77; 

but  /,V(«^  -  ^•^)  =  «^ V(^^^  -  ^)  +/«-77^— Xx' 

62/  integration  by  parts,  if  we  let 

u  =  ^{a^  —  x^)  and  D^v  =  1. 
Adding  our  two  equations,  we  have 

2/,V(«'  -  ^)  =  a;  V(^'  -  a^)  +  a^sin-^? ; 

a 

and  .'.MiaJ'  -  o^)  =  Va^V^FZ^  _^  ^'sin"^^). 


Examples. 
Find 


(1)  M{^  +  a') 


^ris.   -[x^{;x?  +  a')  +  anog{x  +  ^J'^To:')y 


(2)  /.V(«^-«'). 


Applications. 

83.    To  find  the  area  of  a  segment  of  a  circle. 
Let  the  equation  of  the  circle  be 

and  let  the  required  segment  be  cut  off  by  the  double  ordinates 
through  (xo,2/o)  and  {x,y) .     Then  the  requu-ed  area 

A=2f^y  +  C, 


Chap.  V.] 


INTEGRATION. 


73 


From  the  equation  of  the  circle, 

hence  A  =  2/,  V  («'  -  '^•')  +  C ; 

and  therefore,  b}'  Art.  82  (c) , 

A  =  x-^(a^-x')-\-  a'  sin"'  ^*  +  C. 

As  the  area  is  measured  from  the  ordinate  ?/o  to  the  ordinate  ?/, 
A  =  0  when  x  =  ct'o ; 
therefore  0  =  Xo^J(a^-Xo^)  +  a^sin"^-  +  C, 


C=  —Xq  ^{a^  —  Xq^)  —  a^sin"^ — ? 


and  we  haA'e 


A  =  x  V(«^  —  ^)  +  «^sin-^ XQ'^{a^—Xi^^)  —  a^sin"^— 


If  iCo=  0,  and  the  segment  begins  with  the  axis  ofY. 


A  =  x^{a^  —  xr)-{-  a^sin"^- 


If,  at  the  same  time,  a;  =  a,  the  segment  becomes  a  semicircle,  and 


A  =  a-\l{a^  —  a^)  -f  (x^sin~^-  =  ^^ — . 

a       2t 


The  area  of  the  whole  circle  is  7ra^. 


74  DIFFERENTIAL   CALCULUS.  [Art.  84. 

Examples. 

(1)  Show  that,  in  the  case  of  an  ellipse, 

the  area  of  a  segment  beginning  with  an}'  ordinate  t/o  is 

A  =  -  fa;  V(^r  -  x')  +  a^sin-^^  -  x,J(a'-  x,') -  a^sin"^ ^  . 
a\_  a  a  _ 

That  if  the  segment  begins  with  the  minor  axis, 

A=-\ xJ(a^-x')-\-a''^m-'-    . 
a\_  a 

That  the  area  of  the  whole  ellipse  is  -Kob. 

(2)  The  area  of  a  segment  of  the  hyperbola 

is  A  =  -\jc^{^-o?)-anog{x+yf^f~^^) 

(X 

-  a^oV  W-  c^')  +  a21og(a.'o+V^^^2)]. 
If  iCo  =  a,  and  the  segment  begins  at  the  vertex, 

^  =  -  [a;  VCaj"  -  o?)  -  anog(a;  +  V^^^)  +  anoga\ 

Ob 

84.    To  find  the  length  of  any  arc  of  a  circle^  the  coordinates 
of  its  extremities  being  (xo,2/o)  and  {x^y) . 

By  Art.  52,  8=/,V[l+ (^2/)^- 

From  the  equation  of  the  circle. 


Chap.  V.J  INTEGRATION.  75 

we  have  2x-\-2yD^y  =  0^ 

y 


a        ^  1  .  _ia: 

y  -^{or  —  x'^)  a 


s=/^^=a/^-jy-J—^^=asm-'^  +  C.    (Art.  77.) 


When  x  =  Xq^  s  =  0 ; 


Xq 

hence  0  =  a  sin"^  r  +  C', 

u 


(7  =  —  a  Sin  ^  - , 

and  5  =  af  sin~^-  — sin~^-^ 

If  a;o=  0,  and  the  arc  is  measured  from  the  highest  point  of  the 

•     1^ 
circle^  s  =  asin  ^-- 

a 

If  the  arc  is  a  quadrant^     x  =  a, 

•  _i  /I  \      "^a 
s  =  asm  '(1)  =  —  , 

and  the  whole  circumference  =  27ra. 

85.    To  find  the  length  of  an  arc  of  the  parabola  y^  =  2  mx. 
We  have  "^yD^y  =  2 m  ; 

A2/  =  — ; 
y 


76  DIFFERENTIAL   CALCULUS.  [Art.  85. 

D„x  =  J-=y-,  by  Art.  73; 

D,y      m 

y/i  ^  7?!/ 

by  Art.  82,  Ex.  1. 
If  the  arc  is  measured  from  the  vertex, 

s  =  0  when  y  =  0; 

0=— (m2logm)  +  C, 
2m 

(7= mlogm, 

and  « =  1  pV('"^  +  ^^)  +  ,,,       2/  +  V(m-  +  y^)- 

2  [_  m  ^5  ^2, 

Example. 

Find  the  length  of  the  arc  of  the  curve  a^=  27/ included  be- 
tween the  origin  and  the  point  whose  abscissa  is  15. 

Ans.    19. 


Chap.  VI.] 


CURVATURE. 


77 


CHAPTER  VI. 

CURVATURE. 

86.  The  total  curvature  of  an  arc  of  a  continuous  curve  is  its 
total  change  of  direction,  and  is  measured  by  the  angle  formed 
by  the  tangents  at  its  extremities.  The  mean  curvature  of  an 
arc  is  its  total  curvature  divided  by  its  length.  The  actual  curv- 
ature of  a  curve  at  a  given  point  is  the  limit  approached  by  the 
mean  curvature  of  the  arc  beginning  at  the  point,  as  the  length 
of  the  arc  is  indefinitely  decreased. 


Thus,  in  the  figure,  the  total  curvature  of  the  arc  Po  ^^  or  Js,  is 
the  angle  cr,  which  is  equal  to  r  —  tq  or  Jr.    The  mean  curvature 

is  —I,  and  the  actual  curvature  at  Pq  is 
Js 


limit 


=  0  |_J.9j 


87.    To  find  D,t  in  any  particular  example,  we  must,  in  theory, 
begin  by  expressing  r  in  terms  of  s  by  the  aid  of  our  old  relations 


tanr  =  Z)^?/, 


78 


DIFFERENTIAL  CALCULUS. 


[Art.  88. 


together  with  the  equation  of  the  given  curve  ;  but,  in  practice, 
this  part  of  the  work  may  be  avoided.  By  the  aid  of  the  rela- 
tions just  referred  to,  r  and  s  may  be  expressed  in  terms  of  x ; 
and,  consequently,  we  may  regard  them  as  functions  of  a;,  and 
can  obtain  their  derivatives  with  respect  to  x  ;  and  then  the  de- 
rivative of  either  with  respect  to  the  other  may  be  found  by  the 
following  principle. 


88.   If  ?/  is  a  function  of  ic,  and  2  is  a  function  of  x, 


For 


limit 
Jaj=0 


Az 
Ax 
Jy 
Ax 


limit  r^l 
D,z_Ax=0  1Ax]       lijni 

^^y       limit  [^1 
Ax=0  \_Axj 

^  limit  ri!l=  limit  r^1  =  2)  > 
Ax=0lAyj      Ay=0lAy_\ 

for  Ax,  Ay,  and  Az  approach  zero  simultaneously 


[1] 


89.   "We  have  thus,  if  x  represents  the  curvature  at  any  point 


of  the  curve, 
Since 


but 
and 


tanr=  D^y, 


D. 


D.'y. 


sec^r 
sec^r  =  1  -f  tan^r  =  1  +  ( Ay)S 


D.r  = 


Chap.  VI.]  CURVATURE.  79 

and,  as  Z),s  =  V[l  +  (^x2/)'], 


we  have 


D,'y 


±\\  +  {D^yy^i 


Either  the  positive  or  the  negative  value  might  be  chosen  as  the 
normal  one.  For  reasons  that  will  be  evident  hereafter,  it  is 
customary  to  use  the  negative  one  ;  and  we  have 


(a)    For  example,  let  it  be  required  to  find  the  curvature  of  a 
straight  line  Ax -\-By  -\-C=0  at  any  point. 

Differentiating  with  respect  to  x,  we  have 
A-^BD,y  =  0) 

l  +  {D^yy  =  ^^; 
^  _       -  D^^y       _         0 


V     B' 

a  result  which  might  have  been  anticipated. 
(b)    The  curvature  of  a  circle, 

x^-\-y^=  a^. 
2x-\-2yD,y=:0', 

D^y=--', 


80  DITFEEENTIAL  CALCULUS.  [Art.  89. 


y -\ — 

2„.  y-^D,y  y  oi?-\-y'  a? 


f    -- 

f 

f 

l  +  (^.2/)^ 

a" 

5 

1 
a 

Hence  the  curvature  of  a  circle  is  the  same  at  every  point,  and  is 
equal  to  the  reciprocal  of  the  radius. 

If  a  =  l, 

x=l; 

and  the  unit  of  curvature  is  the  curvature  of  the  circle  whose  radius 
is  unity. 

(c)    The  curvature  of  a  parabola, 

y^=2  mx. 

2yD^y=2m', 

D^y  =  —  ; 

y 

2/  f 


i  +  (A2/r=^^^^; 


^2 ' 


'"/   '  f         ~{m'  +  f)V 

and  is  a  function  of  y,  one  of  the  coordinates  of  the  point  con- 


Chap.  VI.]  CURVATURE.  81 

sidered.     From  the  form  of  z,  it  is  obvious  that  the  curvatm-e 

is  greatest  when  y=0\ 

that  is,  at  the  vertex  of  the  curve ;  that  it  decreases  as  y  in- 
creases or  decreases,  and  that  it  has  equal  values  for  values  of 
y  which  are  equal  with  opposite  signs. 

Examples. 
(1)    Required  the  curvature  of  the  ellipse 

-2  +  ^  =  1  at  an}^  point. 


Ans.    y.  = 


o>y 


(b^x'-^-a^y^)^ 


(2)    Of  the  hyperbola     ^-t  =  i. 


(3)    Of  the  equilateral  hjperbola 

a" 
•^       2 


Ans.   X  = 


(6V  +  a'/)^ 


Ans.    z  =  — 


Osculating  Circle. 

90.  As  the  curvature  of  a  circle  has  been  found  to  be  the 
reciprocal  of  its  radius,  a  circle  may  be  drawn  which  shall  have 
any  curvature  required.  A  circle  tangent  to  a  curve  at  any  pointy 
and  having  the  same  curvature  as  the  curve  at  that  poi7it,  is  called 
the  osculating  circle  of  the  curve  at  the  point  in  question.  Its 
radius  is  called  the  radius  of  curvature  of  the  curve  at  the  point, 
and  its  centime  is  called  the  centre  of  curvature. 

From  the  definition  of  the  radius  of  curvature,  it  is  obviously 


82 


DIFFERENTIAL   CALCULUS. 


[Art.  9L 


normal  to  the  curve,  and  its  length  is  the  reciprocal  of  the  curva- 
ture at  the  point.     If  p  represents  the  radius  of  curvature,  we 


have 


P  = 


Of  course,  p  is  generally  a  function  of  the  coordinates  of  the  point 
of  the  curve,  and  changes  its  length  as  the  position  of  the  point 
in  question  is  changed. 

Evolutes. 

91 .    The  locus  of  the  centre  of  curvature  of  a  given  curve  is  the 
evolute  of  the  curve. 

Problem. 


To  find  the  equation  of  the  evolute  of  a  given  curve 

y=fx. 

Let  P,  coordinates  {x^y) ,  be  an}^  point  of  the  curve,  and  P',  {x\y^) 
the  corresponding  point  of  the  evolute ;  v  the  angle  made  by 
the  normal  with  the  axis  of  x^  and  p  the  radius  of  curvature  at 
P.     p  and  T  can  be  found  from  the  equation  of  the  curve,  and 

V  =  r  -  90°. 
We  see  from  the  figure,  that 

x^=x  —  pco^v^ 


y'=y  —  psinv 


Chap.  VI.  ]  CURVATURE.  83 

p  and  V  can  be  expressed  in  terms  of  x  and  y ;  and  then,  with 

the  given  equation,  y=fx-> 

we  shall  have  three  equations  connecting  the  four  variables,  ic,  i/, 
x\  and  y\  "We  can  eliminate  x  and  y,  and  so  obtain  a  single 
equation  connecting  x'  and  y\  the  variable  coordinates  of  any 
point  on  the  e volute  ;  and  this  will  be  the  equation  required. 

92.   For  example :  Let  us  find  the  evolute  of  the  parabola 

2/^=2  mx. 

tanr  =D,y  =  —', 

y 

tanv  =  tan  (r  —  90°)  =  —  cotr  =  —  J^  ; 
sec^v  =  1  +  tan^  v  =  — ^^-^  ; 


COSv=  ± 


m 
m 


sinv  =  ± ^ . 

^(m'  +  f) 

Since  v  is  given  by  its  tangent,  it  may  always  be  taken  less  than 
180°  ;  therefore  we  may  take  the  positive  value  of  sinv,  and  in 
that  case,  as  tanv  is  negative,  we  must  take  cosv  with  the  nega- 
tive sign  :  we  have  then 

V 

V(m2  4-y2) 


We  have  seen.  Art.  89  (c),  that 
_        m^ 


84  DIFFERENTIAL  CALCULUS.  TArt.  93. 

hence  ^  =  (Z!^!+^S 

x'=:x-\ -^-^; 


and  these  equations, 

together  with 

2/^=  2maj, 

are  the  equations  of  the  evolute. 

Reducing,  we  have 

a;'  =  m  +  3a;; 

^      x'-m. 

X-      3      , 

and 

y  =^  — o  > 

whence  2/  =  —  {'m^y')  ^. 

Substituting  in  the  equation  of  the  parabola,  we  have 

o 

Reducing,  m*y'^  =  —  m^(x'—  mY^ 


27m 


or,  dropping  accents, 


21m 


the  required  evolute  ;  a  semi-cubical  parabola. 


Chap.  VI.]  CURVATURE.  85 

93.    By  expressing  p  and  v  in  terms  of  x  and  y  in  the  general 
equations  of  the  evohite  of    2/  =/x, 
we  can  throw  these  equations  into  a  rather  more  convenient  form. 

We  have  the  values    />=_Ii±^^i^\ 


and 


Reducing 


smy 


D.'y 

tanr  = 

=  D.y, 

tani'^ 

1 

C0tj'  = 

-A2/. 

1 

[l  +  CA2/)^]' 


COS»'  =  — 


._,      [l  +  (i>.j/)^3  A?/ 


a?  =  .T 


.y'=  2/  -h 


l  +  (A?/)2 

V  —  y  4 — ^ — '-' '  ■ 


[1] 


Example. 
Required  the  evolute  of  a  circle.  Ans.   a;'=  0,  ?/'=  0. 


94.    To  find  the  evolute  of  an  ellipse^ 


86  DIFFERENTIAL   CALCULUS.  [Art.  94. 


D.y 


a'y 


^  a'y         a'f       \      b'  J~  7i^^  ' 

Since  -t  +  f'=l, 

a*y^  =  (rb^(d^  —  .v-)  ; 
and  b^x'  =  aV)\W-y-). 

or  cr{b'-b^y^-{-ah/). 

^,_^      x{a^-a^x'-^b^x^)_cr-^^,^ 

^    ^  ^4  -       54- r, 

■^         \a'-by 
Substituting  in  -+-^=1, 


Chap.  VI.]  CUKVATURE.  87 

we  have  f_^Y+^^^Y=i; 

or,  dropping  accents, 

Example. 

2  2 

Find  the  evolute  of  the  hyperbola  ^  _  ^  =  j 

Ans.    (aaj)i-(%)l  =  (a2  +  62)l. 


Properties  of  the  Evolute. 

95.  We  have  defined  the  evohite  as  the  locus  of  the  centre  of 
curvature  of  the  curve.  It  is  also  the  envelop  of  the  normals 
of  the  given  curve,  as  may  be  readily  shown ;  that  is,  every 
normal  to  the  curve  is  tangent  to  the  evolute.  Let  v  be  the  in- 
clination of  the  normal  at  the  point  {x^y)  of  the  given  curve  to 
the  axis  of  X,  and  r'  the  inclination  of  the  tangent  at  the  corre- 
sponding point  (x\y')  of  the  evolute.  We  have  seen  already 
that  the  normal  at  {x,y)  passes  through  {x'^y'),  so  it  is  onljf 
necessary  to  prove  that  t'=  v. 

But  tanr'=Z),,;y' 

and  tan  v  =  —  — — . 

D,y 

Hence  we  must  show  that  D^  y'= . 

By  Art.  88,  D^y'=  M, 

since  x'  and  y'  may  both  be  regarded  as  functions  of  a?. 
,       ^._^     D.yl\  +  {D.yy]. 


88  DIFFERENTIAL   CALCULUS.  [Art.  96. 

y=  y + 


,    ..  .  \  +  {D^yY  . 


K'y 
j.^,    1      (D.'yy-\-^D.yY{D.'yY-D.y[_\  +  {D,yy^D.'y 

{D.'yY 

_  3  D^yjDJyY-  [1  +  (D.yYW^'y  . 

WyY 

^^      B^x'  D^y 

96.  A  second  important  property  of  the  evolute  is  that  the 
length  of  any  arc  of  the  evolute  is  the  difference  between  the  lengths 
of  the  radii  of  curvature  of  the  given  curve  which  pass  through  the 
extremities  of  the  arc  in  question. 

Let  {xi\yi')  and  (a'2'^2/2')  be  the  extremities  of  any  arc  of  the 
evohite  ;  pi  and  /?2  the  radii  of  curvature  drawn  from  these  points 
to  the  curve ;  s/  the  length  of  the  arc  of  the  evohite  measured 
from  some  fixed  point  on  the  evohite  to  (a?/,?/]')  ;  and  Sg'  the 
length  of  an  arc  measured  from  the  same  fixed  point  to  (iPg',  2/2')  • 
Then  we  wish  to  prove  that 

S2'—Si=p2  —  Pl, 

or  Js'=Jp, 

Jp 


limit  r^n=i 

ip=0ljp]      ' 
Dps'=l, 
where  s'  must  be  regarded  as  a  function  of />. 


or 

Jp 


or  Dps'=l 


Chap.  VI.]  CURVATURE.  89 

But  DoS'=^ 

and  '  D,s'=  ^^  =  D,.  s'.  D,x'. 

lK,7y= —,  by  Art.  95; 

hence  ^^'^-7^  [l  +  (^.2/)']^. 

{D^yY 
by  Art.  95. 

[l  +  (D..v)^]i. 

X>pS'=^'  =  l.  Q.E.D. 

97.  These  two  properties  enable  us  to  regard  any  curve  as 
traced  by  the  extremity  of  a  stretched  string  unioound  from  the 
evolute,  the  string  being  always  tangent  to  the  evolute,  and  its 
free  portion  at  any  instant  being  the  radius  of  curvature  of  the 
curve  at  the  point  traced  at  that  instant.  From  this  point  of 
view,  the  curve  itself  is  called  the  involute. 


90  DIFFERENTIAL  CALCULUS.  [Art.  98. 


CHAPTER  VII. 

SPECIAL   EXAMPLES   AND  APPLICATIONS. 

The  Cycloid. 

98.  The  cycloid^  a  plane  curve  possessing  very  remarkable 
geometrical  and  mechanical  properties,  was  first  studied  just 
before  the  invention  of  the  Calculus,  and  has  alwaj^s  been  a 
favorite  with  mathematicians. 

It  is  the  curve  described  in  space  by  a  fixed  point  in  the  rim 
of  a  wheel  as  the  wheel  rolls  along  in  a  straight  line  ;  or,  more 
strictly,  it  is  the  curve  described  by  any  fixed  point  in  the  cir- 
cumference of  a  circle,  as  the  circle,  keeping  alwa3's  in  the  same 
plane,  rolls  without  sliding  along  a  fixed  straight  line.  The 
rolling  circle  is  called  the  generating  circle^  and  the  fixed  point 
the  generating  pointy  of  the  cycloid. 

The  curve  will  evidently  consist  of  an  indefinite  number  of 
equal  arches,  and  can  be  cut  by  a  straight  line  in  an  unlimited 
number  of  points.  Its  equation,  then,  cannot  be  of  a  finite 
degree,  and  so  cannot  be  an  algebraic  equation.  The  curve  is 
a  transcendental^  as  distinguished  from  an  algebraic^  curve. 

99.  As  the  arches  are  all  alike,  it  will  do  to  consider  a  single 
one.  Its  base  is  obviously  equal  to  the  circumference,  and  its 
height  to  the  diameter,  of  the  generating  circle,  and  its  right  and 
left  hand  halves  are  S3'mmetrical. 

We  can  get  its  equation  most  easily  with  the  aid  of  an  auxil- 
iary angle.  Take  as  axes  the  base  of  the  cycloid,  and  a  per- 
pendicular to  the  base  through  the  lowest  position  of  the  generat- 
ing point,  and  represent  by  0  the  angle  made  by  the  radius 
drawn  to  the  generating  point  at  any  instant,  with  the  radius 
drawn  to  the  lowest  point  of  the  generating  circle.     The  arc 


Chap.  VII.]    SPECIAL  EXAMPLES   AND   APPLICATIONS. 


91 


joining  the  two  points  just  mentioned  is  aO^  by  Art.  66,  if  a  is 
the  radius  of  the  circle  ;  and  this  is  therefore  the  length  OT.  If 
X  and  y  are  the  coordinates  of  P,  any  point  on  the  cycloid, 


x  =  aO  —  asin^ 
y  =  a  —  a  cos  0 


(^) 


and  these  may  be  taken  as  the  equations  of  the  cycloid.  Of 
course,  0  may  be  eliminated  between  these  equations,  and  a 
single  equation  obtained,  containing  x  and  y  as  the  only  varia- 


bles.    We  get 


cos^  = 


a-y 


1  —  cos  0=-  =  vers  0, 


hence 


0  =  YeYS~^- 


smO  =  ^{l-cos^O)=  ±-^{2ay-y^). 


and 


y 

X  =  avers-^-  q:  ^(2ay  -  /) , 


(B) 


where  the  upper  sign  before  the  radical  is  to  be  used  for  points 
corresponding  to  values  of  ^<7r,  that  is,  for  points  on  the  first 
half  of  the  arch,  and  the  lower  sign  for  points  on  the  second 
half  of  the  curve. 


92 


DIFFERENTIAL  CALCULUS. 


[Art.  100. 


Examples. 

( 1 )  Discuss  completely  the  form  of  the  cj^cloid  from  equations 
(^),  supposing  0  to  increase  from  0  to  2::. 

(2)  Discuss  the  form  of  the  cycloid  from  equation  (i?),  sup- 
posing y  to  increase  from  0  to  2  a. 

100.  If  our  axes  are  lines  parallel  and  perpendicular  to  the 
base  through  the  highest  point  of  the  curve,  the  equations  have 
a  slightly  different  form.  Let  6  be  measured  from  the  highest 
point  of  the  generating  circle. 


and 


OT=AB  =  ad 

x  =  aO-\-  a  sin  0 
y=  —a -{-a  cos  0 


(O) 


( 

^                  r        X 

/-"^'^ 

V 

^ 

V 

p 

/ 

K 

A                  E 

f 

Examples. 

(1)  Obtain  equations  (C)  from  equations  (A)  by  transform- 
ation of  coordinates,  noting  that  the  formulas  required  are 

x=a7:  -{-x\ 

2/  =  2a  +  2/', 

(2)  Eliminate  0  and  obtain  a  single  equation  for  the  cycloid 
referred  to  its  vertex  as  origin. 


Chap.  VII.]    SPECIAL  EXAMPLES   AND  APPLICATIONS.  93 

101.  The  properties  of  the  curve  can  be  investigated  from  the 
equations  involving  0  or  from  the  single  equation.  In  the  text 
we  shall  employ  the  former.  We  ought  to  be  able  to  determine 
(1)  the  direction  of  the  tangent  and  normal  at  any  point  of  the 
curve  ;  (2)  the  equations  of  tangent  and  normal ;  (3)  the  lengths 
of  tangent,  normal,  subtangent,  and  subnormal ;  (4)  the  curva- 
ture of  the  cycloid  at  any  point ;  (5)  the  evolute  ;  (6)  the  length 
of  an  arc  of  the  curve  ;   (7)  the  area  of  a  segment  of  the  curve. 

(1)    102.  x  =  aO  —  asine, 

y  =  a  —  acosd, 

r.         D.y         asinfi  sin^         2sin|cos| 

tanr  =  D^y  =  —^  = = = ^  =  cotg ; 

I)qX      a  —  aeosd      1— cosy         2sin^| 

tanv  =  —  cotr  =  —  tan|. 

Since,  as  we  have  seen  in  Art.  99, 

smd=-^J{2ay-y^) 


and 

1— cosy  =  ^, 
a 

tanr  can  be  written 

=>!(?-> 

anfi 

f  on  ij  ^ 

UIIU. 

*^"'-       V(2«2/-/) 

Since 

tanv/=  —  tan|, 

.=.-1 

by  trigonometry. 

In  the  figure  (see  next  page) ,  PTO  being  formed  by  a  tangent 
and  a  chord,  is  measured  by  half  the  arc  PT,  and  therefore  is 
equal  to  |.  P7M,  then,  is  equal  to  v,  and  the  line  PT  is  a  normal. 
Hence  the  normal  at  any  point  on  the  cycloid  passes  through  the 
lowest  point  of  the  generating  circle.  The  tangent  must,  therefore, 
pass  through  the  highest  point  of  the  generating  circle. 


94 


DIFFERENTIAL  CALCULUS. 


[Art.  103. 


Examples. 

(1)  At  what  point  of  the  curve  is  the  tangent  parallel  to  the 
base  of  the  cycloid  ?  Perpendicular  to  the  base  ?  Where  does 
it  make  an  angle  of  45°  with  the  base  ? 

(2)  Obtain  the  values  of  tan  r  and  tanv  from  equation  (B). 


(2)    103.   The  equation  of  the  tangent  at  the  point  {xQ.yo)  is 
by  Art.  28,  y  —  yQ=cot^  (x  —  Xq) , 

or  y-yQ=lf^-i\(x-Xo)  ; 

of  the  normal,  is      y  —  yQ=  —  tan  2^  {x  —  Xq)  , 


or 


y-yo=- 


V(2«2/o-2/o^) 


{x  -  Xo) . 


Example. 

Show,  from  the  equation  of  the  normal,  that  it  passes  through 
the  point  (a^,0),  the  lowest  point  of  the  generating  circle. 


(3)    104.   We  have  the  formulas, 

t  -    ^ 

n^  =  yDxy, 


Chap.  VII.]   SPECIAL  EXAMPLES   AND   APPLICATIONS.  95 

n  =  2/V[l  +  (A2/)'];  (Art.  28); 

a(l  — cos^A      2asin2^ 

hence  t,  =  -^ i  = —  =  2  a  sin^ltanj  • 

cot!j  cot  I 

w^=rt(l-cos<?)cot|=2asin2§cot.f  =  2asin|cos|=asin^, 
t=2a  sin^ftanl  cscf  =  2  «  sinf  tanf. 
11=  2asin^|csc|=  2«sin|. 
Since  />.2,  =  ^(!£'_l), 

the  value  of  n  may  be  expressed, 

^e^  2osin2|  4a 

[l  +  (A.y)^^  =  csci, 

hence  x=  — po^.e 

and  /o  =  -  =  4asin|=2>i=2V(2«?/)  ; 

and  the  radius  of  curvature  at  any  point  is  equal  to  twice  the 
normal  drawn  at  the  point. 


96 


DIFFERENTIAL   CALCULUS. 


CArt.  106. 


Examples. 

(1)  Find  at  what  points  of  the  curve  the  curvature  is  great- 
est ;  at  what  least. 

(2)  Obtain  the  expression  for  the  curvature  from  the  equa- 
tion (B). 

(5)    106.   The  equations  of  the  evolute  of  a  curve  are 


DxV 


y  =  y-h 


(Art.  93  [1]). 


Here 


cot|csc^§ 
x'=ad-asm0 ^  =  a^  — asin^  + 4asin|cos| 

-J^csci 


=  a^  — asin^  +  2asin^, 
=  a^  +  asin^^. 


2/'=a  — acosi^^- 


csc^l 


r~  CSC"*! 
4  a        2 


a  —  a  cos  0  —  4:a  sin^§ 


=  o(l  — cos^)— 2a(l—  cos^)=  —  a  +  acos^; 
and  we  have,  as  the  equations  of  the  evolute, 
flj'=a^  +  asin^  ] 
y'=  —a  +  acoso] 

but  these  (Art.  100)  are  the  equations  of  an  equal  cycloid  re- 
ferred to  the  tangent  and  normal  at  the  vertex  as  axes.     The 


Chap.  VII.]   SPECIAL  EXAMPLES   AND  APPLICATIONS.         97 

cycloid  and  its  evolute  would  be  situated  as  indicated  by  the 
figure. 

The  property  of  the  evolute  established  in  Art.  96  enables  us 
to  obtain  easily  the  length  of  the  arc  of  an  arch  of  the  cycloid 


The  length  of  the  half-arch  of  the  evolute  is  the  difference  be- 
tween the  radii  of  curvature  at  the  highest  and  the  lowest  points 
of  the  given  curve  ;  that  is, 

[/>]0=ff—  [/>]0=o=  4  a  sin-  —  4asin0  =  4a, 

and  /S',  the  whole  arc,=  8a. 


(6)    107.   The  length  of  an  arc  of  the  cycloid  can  be  found 

from  the  formula  s  =/x[l  +  (A^)^]^ 

without  using  the  evolute. 

We  have  D^y  =  cot^, 

hence  s  =  /^  esc  |  =/g  esc  \D(^X'. 

but  DQX=2a^m% 

and  s=2a/gsin|. 


98  DIFFERENTIAL   CALCULUS.  [Art.  108. 

Let  z  =  l; 

then  DJ  =  2, 

s  =  2  a/g  siii2  =  2  a/^  smzD^O=  —  4  a  cos  2;  +  (7, 
s=  —  4  a  cos  ^  +  (7. 
If  we  measure  the  arc  from  the  origin,  s  must  equal  0  when 

0=  -4acos0  +  (7, 
C=4a, 
and  we  have  s  =  4a(l  — cos|). 

To  get  the  whole  arch,  let    ^  =  2?:, 

s  =  4  a  ( 1  —  cos ;:)  =  8  a. 

(7)    108.   For  the  area  of  a  segment  of  the  arch,  we  have  the 
formula  A=f^y  +  C. 

fxV  =  cifxC^ -  cos 0)  =  a/g(l- cos 0)DQX=a%{l- cos oy 
=  a%{l-  2cosd -\-  cos'0)  =  a'(fQl-2/QCOsO  +fgCos^e), 

fg  COS  0  =  sin  0, 
Sq  cos^  ^  =  ^  (  ^  +  sin  ^  cos  0) 
[see  Art.  80,  Ex.  (1)]  ; 

hence        ^  =  a2[^  — 2sin6'  +  i(^-|- sin^cos^)]  +  C. 
If  the  segment  is  measured  from  the  origin, 
^  =  0  when  ^  =  0  ; 
0  =  a2[0-0  +  ^(0  +  0)]  +  (7 
and  C=0. 


Chap.  VII.]    SPECIAL  EXAJVIPLES   AND   APPLICATIONS.  99 

The  area  of  the  whole  arch  is  obtained  by  making 

^  =  a2[2--2sin2-+ i(2  7:  +  sin2-cos2  7r)]  =  3  7ra2, 

so  that  the  area  of  the  arch  is  three  times  the  area  of  the  gen- 
erating circle. 

Example. 

Find  the  length  of  an  arc  and  the  area  of  a  segment  from  the 
equation  (J5). 

109.  If  the  generating  circle  rolls  on  the  circumference  of  a 
fixed  circle^  instead  of  on  a  fixed  line,  the  curve  generated  is 
called  an  epicycloid^  if  the  rolling  circle  and  the  fixed  circle  are 
tangent  externally^  a  hypocycloid,  if  they  are  tangent  internally. 
The  equations  of  these  curves  may  be  readily  obtained.    Let  the 


figure  represent  the  generation  of  an  epicj'cloid,  P  being  the 
generating  point  and  E  the  starting  point.  Call  AOB,  0  ;  and 
PCA^  <f  ;  OD  is  x  and  DP  is  y.  Let  a  and  h  be  the  radii  of 
fixed  and  rolling  circles.     Then 


100 


DIFFERENTIAL   CALCULUS. 


[Art.  109. 


ic  =  (a 4- 6) cos ^  4- 6 sin   <p—l-—o\  L 
y  =  (a-\-b)smO  —  bcos\  ^— (^  — ^)    ; 


but  the  arcs  AP  and  AE  are  equal,  and 

AP=b<p, 
AE  =  aO, 

hence  ad  =  hep 


and 


0 


and  the  equations  become 


a  +  6 


cc  =  (a  -f  6)  cos  0  —  h  cos  ^ 


y  =  (a  +  &)sin^-  &sin^^-±-^  0 
h 


[1] 


The  equations  of  the  h}^ocycloid  are,  in  like  manner,  found  to 
be 


x  —  {ci  —  b)  cos  0  -{-b  cos o 

b 


y  =  {a-  &)sin^  -  6sin^^^ — ^  0 
b 


Examples. 


[2] 


(1)    If  6  =  a  in  the  epicj^cloid,  the  curve  is  called  a  cardioide. 
Show  that  its  polar  equation  is 

r=2a(l—  coscp) 
when  the  starting  point  is  taken  as  pole. 


Chap.  VII.]   SPECIAL  EXAMPLES  AND  APPLICATIONS.       101 

(2)  Ifa  =  45in  the  hypoc3'cloid,  obtain  the  cartesian  equa- 
tion of  the  curve  by  eliminating  0.  Ans.   x^-{-y^  =  a^. 

(3)  If  a  =  26  in  the  hypocyloid,  show  that  the  curve  reduces 
to  a  diameter  of  the  fixed  circle. 

(4)  Prove  by  diiferentiation  that  the  normal  at  any  point  of 
either  epicj'cloid  or  hypoc3'cloid  passes  through  the  point  of  con- 
tact of  fixed  and  generating  circles. 


102  DIFFERENTIAL  CALCULUS.  [Art.  110. 


CHAPTER  VIII. 

PROBLEMS   IN  MECHANICS. 

110.  We  have  seen  (Art.  12)  that,  if  s  represents  the  dis- 
tance traversed  by  a  moving  body  in  t  seconds,  and  can  be 
expressed  as  a  function  of  t,  the  velocity  of  the  body  at  any 

instant  v  =  DtS. 

111.  The  acceleration  of  a  moving  body  at  any  instant  is  the 
rate  at  which  its  velocity  is  changing  at  that  instant.  If  the 
velocity  is  increasing,  the  acceleration  is  positive ;  if  diminish- 
ing, the  acceleration  is  negative.  We  shall  represent  it  by  a, 
and  it  is  evidently  a  function  of  t.  Since  the  derivative  of  a 
function  measures  the  rate  at  which  its  value  is  changing  (Art. 

38) ,  we  shall  have  a=DtV=  D^s^ 

since  v—DtS. 

For  example :  in  the  case  of  a  body  falling  freely  near  the  sur- 
face of  the  earth,  we  have  approximately^  the  law 

s  =  16^^ 

Here  'y  =  Z>,s=32^ 

and  a=D,v=D?s  =  ^'l, 

and  the  acceleration  is  constant  and  is  equal  to  32  feet  a  second ; 
that  is,  the  velocity  of  the  fall  at  any  instant  is  32  feet  a  second 
greater  than  it  was  a  second  before.     The  relations 

and  a=DtV=DtS^ 


Chap.  VIII.]  PROBLEMS  IN  MECHANICS.  103 

and  the  corresponding  formulas, 

obtained  by  integrating  them,  are  of  great  importance  in  prob- 
lems concerning  motion. 

112.  We  shall  assume  the  following  principles  of  mechanics  : 

(1)  A  force  acting  on  a  body  in  the  line  of  its  motion  produces 
an  acceleration  proportional  to  the  intensity  of  the  force  ;  and  this 
acceleration  is  taken  as  the  measure  of  the  force.  We  speak  of 
a  force  as  a  force  producing  an  acceleration  of  so  many  feet  a 
second;  or,  more  briefly,  as  a  force  of  so  many  feet  a  second. 

(2)  The  effect  of  a  force  in  producing  acceleration  in  any  direc- 
tion not  its  own^  is  the  product  of  the  magnitude  of  the  force  by 
the  cosine  of  the  angle  between  the  two  directions;  or,  in  other 
words,  it  is  the  projection  of  the  line  representing  the  force  in 
direction  and  intensity  upon  the  line  of  the  direction  in  question. 

Problem. 

113.  The  force  exerted  by  the  earth's  attraction  upon  any 
particle  of  matter  is  constant  at  any  given  part  of  the  earth's 
surface,  and  is  nearly  equal  to  32  feet  a  second.  Let  g  repre- 
sent the  exact  value  of  this  force  at  any  given  point  of  the 
earth's  surface,  required  the  velocity  of  a  falling  body  at  the 
end  of  t  seconds,  and  the  distance  fallen  in  t  seconds.  Here  a 
is  constant  and  equal  to  g. 

If  the  body  falls  from  rest,  its  velocity  is  0  when  Hs  0 ; 
0  =  (7XOH-0, 
C=0; 
anc.  v  =  gt. 


104  DIFFERENTIAL   CALCULUS.  [ART.  114. 

s=ftV=f,gt  =  gf,t  =  ^gf^-C. 

When  Hs  0,  the  distance  fallen  must  be  0 ; 
0  =  x^x0  +  (7, 

(7=0, 

and  s  =  ^gt^. 

If  the  body,  instead  of  being  dropped,  had  started  with  an  initial 
velocity  Vq, — for  example,  if  it  had  been  fired  from  a  gun  directly 
down  or  directly  up, — we  should  have  found  a  different  value 
for  G  in  the  expression  for  the  velocity, 


v  =  gt  +  G', 

for  now 

,  when 

i  =  0, 

^  =  ^0; 

hence 

Vo  =  f/XO  +  C', 

C=v,, 

and 

v  =  gt-\-Vo. 

S: 

=y;^= 

--/t{9t  +  Vo)  =  i9t'-hVot  +  C; 

but  as 

s=0when^  =  0, 
(7=0, 

and 

s=igt^-{-VoL 

114. 

The 

equation 

a  =  g 

or 

D,'s  =  g 

can  be  integrated  by  a  second  method  of  considerable  interest 
and  generahty.     Multiply  both  members  by  2i)<s. 

2DtsD,^s=2gD,s; 


Chap.  VIII.]  PROBLEMS  IN  MECHANICS.  105 

but  2D,sD^s  =  D,{D,sy\ 

hence  f,2D,sD^s  =  (D,sy, 

and  we  have  (DtsY  =  2gJ]DfS  =  2gs  -\-C 

or  v-=2gs-\-C. 

In  the  case  of  a  falling  body,  when 

^  =  0,  V  =  0,  and  s  =  0  ; 

hence  C  =  0 

and  v^=2gs, 

v  =  ^{2gs),  '  [1] 

or  D,s  =  ^{2gs), 

We  cannot  integrate  directly  here,  for  the  first  member  is  a 
function  of  t  and  the  second  member  a  function  of  s  ;  but  since 

As  =  -^,  by  Art.  73, 

DJ  =  — - —  = — —  s-h. 
^{2gs)       V(2^) 

V(2{/)  -Ji^g)  MKgJ 

Since  •  s  =  0  when  t  =  0, 

C=0 


and  t 


4j} 


It  is  easily  seen  that  these  new  values  for  v  and  t  are  entirely 
consistent  with  those  obtained  in  the  last  article. 


106  DIFFERENTIAL  CALCULUS.  [Art.  115. 

115.    If  the  force  is  any  constant  force  /,  instead  of  g^  we 
S(  have  merely  to  substitute  /  for  g  in  the  preceding 
results.     For  example,  take  the  case  of  a  body 
sliding  without  friction  down  an  inclined  plane. 
Here,  by  Art.  112,   (2),  the  accelerating  force 
in  the  direction  of  the  motion  is  gcos  (90°  —  <p) ,  therefore 

a  =  ^sin^, 

v  =  '^{2gsin<p.s), 

2s 


and  t 


=x- 


sin  <p 


when  there  is  no  initial  velocity.  In  this  case,  the  velocity  and 
time  are  easily  expressed  in  terms  of  the  vertical  distance 
through  which  the  body  has  descended.  Let  OP  be  s,  and  OA^ 
the  vertical  distance,  be  y.     Then 

y  =  ssm(p, 

\Vf/ssmv7      \\gyj 

Substitute  y  for  s  in  Art.  114,  [1]  and  [2],  and  we  get,  as  the 
velocity  the  body  would  acquire  falling  freely  through  the  verti- 
cal distance  i/,  and  the  time  required  for  the  fall, 

v  =  ^{2gy), 
''2y^ 


->!&' 


We  see  the  two  velocities  are  identical ;  that  is,  the  velocity 
acquired  by  a  body  descending  an  inclined  plane  is  precisely 
what  it  would  have  acquired  falling  through  the  vertical  distance 
it  has  actually  descended. 


Chap.  VIII.J  PROBLEMS  IN  MECHANICS.  107 

-  is  the  mean  velocity  of  the  body  during  its  descent,  and 
z 

for  the  inclined  plane, 


=    I  (^  for  the  falling  body. 


t 

Hence  the  mean  velocity  of  a  body  descending  an  inclined  plane 
is  equal  to  the  mean  velocity  of  a  body  lohich  has  fallen  freely  the 
same  vertical  distance. 


116.   Let  the  figure  represent  a  vertical  circle.     The  time  of 
descent  of  a  body  sliding  down  an}-  chord  is 


'4m-4C-T')-4. 


2ssec{d0°-<p) 
9 


by  Art.  115.     If  a  is  the  radius, 

s.sec(90°-^)  =  2a 


and  t  =  2.  ""^ 


9^ 

which  is  also  the  time  a  body  would  require  to  fall  vertically  the 
distance  2  a.  Therefore,  the  time  of  descent  down  a  chord  of  a 
vertical  circle  from  the  highest  point  of  the  circle  to  any  point 
of  the  circumference  is  constant,  and  is  equal  to  the  time  it  would 
take  the  body  to  fall  from  the  highest  to  the  lowest  point  of  the 
same  circle. 


108  DIFFERENTIAL   CALCULUS.  [Art.  117. 

Example. 

Show  that  the  time  of  descent  down  a  chord  from  any  point 
of  a  vertical  circle  to  the  lowest  point  of  the  circle  is  constant. 

Problem. 

117.   To  find  the  velocity  acquired  by  a  bod}'  falling  from  a 
„  distance  toward  the  earth  under  the  influence  of 

-^  0 

the  earth's  attraction. 

Here  we  cannot  regard  the  attracting  force  as 
constant,  as  we  do  in  dealing  with  small  distances 
near  the  surface  of  the  earth,  but  must  take  it  as 
p  inversely  proportional  to  the  square  of  the  distance 

of  the  bod}'  from  the  centre  of  the  earth.  Let  R 
be  the  radms  of  the  earth ;  Tq  the  distance  from 
the  centre  of  the  earth  to  the  point  at  which  the 
body  started  ;  r  the  distance  from  the  centre  to 
the  position  of  the  falling  body  when  the  time  t 
has  elapsed.  Let  g  be  the  force  of  the  attraction 
of  the  earth  at  the  earth's  surface,  and  /  the  force 
exerted  at  P. 

f     R^ 
Then  we  have  -^  z=  — 


or  f=^--  =  a. 


s,  the  distance  fallen  in  the  time  f ,  equals  ?o  —  r. 
hence  —  D^  r  =  ?-—, 

IT 


Chap.  VIII.]  PROBLEMS  IN  MECHANICS.  109 

Multiply  hy  2 D,r;  2 D,r  D?r  =  ~^^^'^^^. 

Integrate : 

(D,ry=-2gR'f,^=-2gRV^^=^  +  C; 

and  v^^^  +  C. 


When  the  bod}'  was  on  the  point  of  starting,  its  velocity  was 
zero  ;  hence,  when  ^  =  ^'o?  '^  =  0  ; 

and  0  =  ^+C, 

and  v'=2gE^ 


,2/1       1 


r     r. 


When  the  body  reaches  the  surface  of  the  earth, 

r  =  R 


K      rj 


and      ■  v'=2gR' 


The  greater  the  value  of  Tq  in  this  result  —  that  is,  the  greater 
the  distance  of  the  starting  point  from  the  centre  of  the  earth  — 

the  nearer  —  comes  to  the  value  0,  and  the  nearer  v^  approaches 

to  zMzl.  or  to  2gR.     In  other  words,  the  Hmiting  value  of  the 

R 
velocity  acquired  by  a  body  falling  from  a  distance  to  the  surface 
of  the  earth  under  the  influence  of  the  earth's  attraction,  as  the 
distance  of  the  starting  point  is  indefinitely  increased,  is  y/{2gR) . 
Let  us  compute  roughly  the  numerical  value  of  this  expression. 
g  is  about  32  feet  per  second ;  and  as  we  use  the  foot  as  a  unit 
in  one  of  our  values,  we  must  in  all :  therefore  R  must  be  ex- 


110  DIFFERENTIAL   CALCULUS.  [Atit.  117. 

pressed  in  feet.     E  is  about  4,000  miles,  or  21,120,000  feet. 

V(i2)  =  4,600,  nearly. 

V(2^)  =  V(64)  =  8. 

'^(2gR)  =  36,800  feet,  or  nearly  seven  miles  ;  and  our  required 
velocity  is  nearly  seven  miles  a  second ;  and  neglecting  the  re- 
sistance of  the  air,  this  is  the  velocity  with  which  a  projectile 
would  have  to  be  thrown  from  the  surface  of  the  earth  to  prevent 
its  returning. 

We  can  easily  go  on  and  get  an  expression  for  the  time  of  the 
(all  by  a  second  integration. 

We  have     {D,ry=2gR'(^  -  ^\=  2gR^^^~'^^ ; 
\r      rj  Tor 

-'=^ife)^■^l(^)-<'•      ■ 


V(^)~ 


^{r^r-i^)  ' 


an  expression  to  which  we  can  apply  the  method  of  integration 
by  parts. 


Let 

u  =  r, 

then 

D,u=l; 

and  Ipt 

Hi'—           ^ 

"        VOv-r^)' 

then 

V  =  vers-^?I         by  Art.  77,  Ex.  (2) 

^0 

by  Art. 

79, 

.[1]. 

i2r      r           i2r 
— —  =  rvers  ^ /,vers  ^  — 

Chap.  VIII.J  PKOBLEMS   IN  MECHANICS.  Ill 

Let  z=^, 

then  D,r='^. 

2 

/,vers-i^  =  V, vers-^2  =  ^«  [(2;  -  1)  yers-^z  +  ^{2z-  z")-] 
by  Art.  75,   [1],  and  Art.  81,  Ex.   (3).     Replacing  z  by  its 
value,      /,  vers-i  ^  =  f'^  -  5*^  vers-^^  +  Jir^r  -  r") . 

When  ^  =  »'o?  ^  =  0  ; 

Examples. 

(1)  The  mean  distance  of  the  moon  from  the  earth  being 
237,000  miles,  find  the  velocity  a  body  would  acquire,  and  the 
time  it  would  occupy,  in  falling  from  the  moon  to  the  earth's 
surface,  neglecting  the  retarding  effect  of  the  moon's  attraction. 

(2)  The  force  of  the  sun's  attraction  at  its  own  surface  is 
905.5  feet;  find  the  velocity  a  body  would  acquire,  and  the 
time  it  would  occupy,  in  falling  from  the  earth  to  the  sun. 
Earth's  mean  distance  =  92,000,000  miles ;  sun's  diameter  = 
860,000  miles. 


112 


DIFFERENTIAL   CALCULUS. 


[Art.  118. 


(3)  Find  the  limit  of  the  velocity  a  body  could  acquire  fall- 
ing from  a  distance  to  the  sun. 

(4)  How  long  would  it   take    Saturn   to   fall   to   the   sun, 
Saturn's  mean  distance  being  about  880,000,000  miles? 


Problem. 

118.  To  find  the  velocity  acquired  under  the  influence  of 
gravit}'  by  a  body  sliding  without  friction  down  a  given  curve, 
or  in  an}'  way  constrained  to  move  in  a  fixed  curve. 

Here  the  effective  accelerating  force  is  always  tangent  to  the 
curve  at  the  point  the  moving  particle  has  reached.     Suppose 


the  origin  of  coordinates  at  the  starting  point,  and  let  the  direc- 
tion downward  be  the  positive  direction  of  the  ordinates.  Of 
course,  this  will  amount  to  changing  the  sign  of  D^y  \  that  is, 
will  make  r  the  supplement  of  its  usual  value.     The  acceleration 

a  =  ^cos^  =  ^cos  (90°  —  r)  =  grsinr. 

1)^2/ =  tan  r, 

H-(A2/)'=sec2r, 


1  +  (A2/)' 
A?/ 


[i+{D^yyy. 


=  COS^r, 


smr. 


Chap.  Vlll.]  PROBLEMS   IN  MECHANICS.  113 

hence  a=Z>/s=         ^    *^ 


but  [l  +  (Z>.7/)2]i=i),s; 

D?s  =  g^  =  cjD,y. 
Multiply  by  ^D^s: 

2D,sD,'s=2gD,ijD,s=2gD,y. 

Integrate  with  respect  to  t,  and 

v'  =  (D,sy=2gy-\-a 
If  the  particle  started  from  rest  at  0, 

v  =  0  when  ?/  =  0, 
and  C=0, 

but  this  is  precisely  the  velocity  it  would  have  acquired  in  faUing 
freely  through  the  vertical  distance  y  (Art.  114,  [1]) .  So  we  are 
led  to  the  remarkable  result,  that  the  velocity  of  a  material  par- 
ticle, sliding  without  friction  down  a  curve,  under  the  Influence 
of  gravity,  is  the  same  at  any  instant  as  if  it  had  fallen  freely  to 
the  same  vertical  distance  below  the  starting  point.  A  special 
case  of  this  has  already  been  noticed  in  Art.  115. 

Example. 

Prove,  from  the  equation  of  a  circle,  and  the  equation  of  a 
chord  through  its  highest  point,  that  the  time  of  descent  is  inde- 
pendent of  the  length  of  the  chord.        ^ 

Problem. 

119.  To  find  the  time  of  descent  of  a  particle  from  any  point 
of  the  arc  of  an  inverted  cycloid  to  the  vertex  of  the  curve. 
Taking  the  origin  at  the  vertex  of  the  curve,  its  equations  are 


114 


DIFFERENTIAL   CALCULUS. 


[Akt.  119. 


x=aO-\-a  sin  0 
y  =  a  —  a  cos  0 


(Art.  100). 


Let  yn  be  the  ordinate  of  the  starting  point,  and  y  the  ordinate 
of  the  point  reached  after  t  seconds.     Then  the  vertical  distance 


fallen  isyo-y,  and        v  =  ^2g{yQ-y), 


by  Art.  118. 


1 


-DJ  = 


t=/s 


1  r  1 


fy 


V2^(2/o-2/)      '    V2gr(2/o-2/) 


^.s; 


As  =  Vi  +  (A^)^ 


DqX^ci  -f  acos^; 

Z>02/  =  asin(9; 

n  ^_  A«_  l±cos^. 
Dey        sm6' 


cos  (9  = 


«-2/ 


Chap.  VIII.]  PROBLEMS   IN   MECHANICS.  115 

2a-2/. 


sin2^  =  l-~cosVy 


a 

a^  —  a^  -\^2ay  —  y^      2ay  — 


a' 


sin^y  =  -V(2«?/-2/^); 


Dcc=       ^""-y       =^  {2a-yy^     pa-y\ 
'         ^/{2ay-f)       \  {2  a -y)y      >JV      2/     7 


by  Art.  77  (2) .     When    ?/  =  ?/o,  i  =  0  ; 


vers-i^  +  C, 


hence  0  =  Jf -jvers-i(2) +0. 

vers~^(2)  is  the  angle  which  has  the  cosine  —1,  that  is,  the 
angle  tt.     Hence,  C=  —    \(-\  tt, 

and  _^  =  J^«Vvers-^ 


.9/  V  2/0 

When  the  particle  reaches  the  vertex, 

2/=0, 

vers"^  — =  0, 

2/0 

and 


•) 


=-ii) 


116  DIFFERENTIAL   CALCULUS.  [Art.  120. 

As  this  expression  is  independeiit  o/yo?  t^^^  ordinate  of  the  starting 
pointy  the  time  of  desceyit  to  the  vertex  will  he  the  same  for  all 
points  of  the  curve.     If  a  pendulum  were  made  to  swing  in  a 

cycloid,  this  time  ^ \i[-]  would  be  one-half  the  time  of  a  com- 
plete vibration,  which  would  therefore  be  independent  of  the 
length  o*f  the  arc.  On  account  of  this  property ,  the  cycloid  is 
called  the  tautochrone  curve. 


Example. 

120.  It  is  shown  in  mechanics,  that,  if  the  earth  were  a  per- 
fect and  homogeneous  sphere,  and  a  cylindrical  hole  having  its 
axis  coincident  with  a  diameter  were  bored  through  it,  the  at- 
traction exerted  on  any  body  within  this  opening  would  be  pro- 
portional to  its  distance  from  the  centre.  Find  the  expression 
for  the  velocity  of  a  body  at  any  instant,  supposing  it  to  have 
been  dropped  into  this  hole,  and  the  time  it  would  take  to  reach 
any  given  point  of  its  course.  Compute  (1)  its  velocit}'  when 
half-way  to  the  centre  ;  (2)  when  at  the  centre  ;  (3)  the  time  it 
would  take  it  to  reach  the  centre,  if  dropped  from  the  surface ; 
(4)  if  dropped  from  an}'  point  below  the  surface.  Given  g  =  S2  ; 
i2,  the  radius  of  the  earth,  =  4,000  miles. 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  117 


CHAPTER   IX. 

DEVELOPMENT   IN   SERIES. 

121.  A  series  is  a  sura  composed  of  an  unlimited  number  of 
terms  which  follow  one  another  according  to  some  law.  If  the 
terms  of  a  series  are  real  and  finite,  the  sum  of  the  first  n  terms 
is  a  definite  value,  no  matter  how  great  the  value  of  n.  If  this 
sum  approaches  a  definite  limit  as  n  is  indefinitely  increased^  the 
series  is  convergent;  ifnot^  it  is  divergent.  The  limit  approached 
by  the  sum  of  the  first  n  terms  of  a  convergent  series  as  n  in- 
creases indefinitely,  is  called  the  sum  of  the  series^  or  simply 
the  series.     Thus,  we  may  express  the  result  arrived  at  in  Art.  6 

by  saying  the  sum  of  the  series  1+^  +  ^  +  8+ is  2;  or, 

more  briefly,  the  series  l-\--^-\-l-{-l+ =  2. 

Example. 

122.  Take  the  series  1  -f  a;  -f-  x^  -f-  ar'  + ,  ad  infinitum.    The 

series  is  a  geometrical  progression^  and  the  sum  of  ?i  terms  can 

be  found  by  the  formula     s= . 

•^  r-1 

«"  —  1       1  —  JC" 


Here 


x-1       1 


and  the  sum  of  the  series  = ,  a  definite  value,  and  the  series 

1  —  x 

is,  therefore,  convergent. 


118  DIFFERENTIAL  CALCULUS.  [Art.  123. 

If  a;>  1 ,  a;"  =  00  when  n=  cc  ^ 

and  the  sum  increases  without  limit  as  the  number  of  terms  in- 
creases indefinitely,   and  the  series  is  divergent.     The  series 

\-\-x-\-x^-\-x^-\- can  be  obtained  from by  actual  divi- 

\  —  x 

sion,  hut  the  fraction  and  the  series  are  equal  only  ivhen  x<l  ; 

for has  a  definite  value  when  x>l  ;  but,  as  we  have  seen, 

1  —  x 

the  series  in  that  case  has  not  a  definite  sum.  It  is  very  unsafe 
to  make  use  of  divergent  series^  or  to  base  any  reasoning  upon 
them,  for,  from  their  nature,  they  are  wholly  indefinite.  Con- 
vergent series,  on  the  other  hand,  are  perfectly  definite  values. 

It  is  easil}'  seen  that  the  sum  of  the  first  n  terms  of  a  series 
cannot  approach  indefinitely  a  fixed  value  as  7i  is  increased,  un- 
less, as  we  advance^in  the  series,  the  terms  eventually  decrease; 
or,  in  other  words,  unless  the  ratio  of  the  nth  tenn  to  the  one 
before  it  eventually  becomes  and  remains  less  than  unity  as  n  is 
increased.  This,  however,  affords  only  a  negative  test  for  the 
convergency  of  series,  as  a  series  may  not  be  convergent  even 
when  each  term  is  less  than  the  term  before  it. 

123.  The  series  we  have  just  considered  is  an  example  of  a 
series  arranged  according  to  the  ascending  powers  of  a  variable, 
and  such  series  play  an  important  part  in  the  theor}-  of  functions. 
We  are  naturally  led  to  the  consideration  of  terms  of  such  a 
series  whenever  we  attempt  to  obtain  a  function  from  one  of  its 

derivatives.     Suppose    Dj^f{xQ  -\-h)  =  z 


where  h  is  a  variable,  Xq  a  given  value,  and  z,  of  course,  a  func- 
tion of  h.     Let /2  stand  for  /,  &c. ,  so  that /"  =Jf''-\ 


Then  nrVi^o  4-  h)  =A,  +f,z, 

where  A^  is  a  constant ; 

Dr'f{^o+h)=A,  +  A,h+f^'z, 


Chap.  IX.]  DEVELOPMENT  IN   SERIES.  119 

DrVi^o  +  h)  =A,  -{-A,h  H-  ^A,h'  +  ^A,h'  +/,% 


f(xo  4-  h)  =A„  +  A-i/i  +  ^  A„_,h'  +  ^  A-3/i'  + 

H '^ A,h''-'  +7/2, 

^2.3 (n-1)     '  -^^    ' 

and  we  have  a  set  of  terms  arranged  according  to  the  ascending 
powers  of  h.  Although,  by  increasing  n  indefinitely,  we  can 
make  the  second  member  above  a  true  series,  it  does  not  by 
any  means  follow  that  every  function  can  be  developed  into  such 
a  series.  In  the  first  place,  it  may  not  be  possible  to  increase 
71  indefinitely  in  the  expression  above,  as  the  nth  derivative  of 
the  function  may  become  at  last  infinite  or  discontinuous,  so 
that// 2  cannot  be  dealt  with.  Next,  the  series  may  be  a  diver- 
gent series,  and  then  it  could  not  be  equal  to  the  definite  value 
f{xQ-\-h).  But  the  result  is  a  remarkable  one,  and  suggests 
the  careful  investigation  of  the  development  of  functions  in 
series. 

124.  Assuming,  for  the  moment,  that/(a:o  +  h)  can  be  devel- 
oped into  a  convergent  series  arranged  according  to  the  ascend- 
ing powers  of  h,  let  us  see  what  the  coeflficients  of  the  series 
must  be.     Let 


f(xo-^h)=Ao-hA,h+A,h'-hA,h'  + +AJi^  + 


The  function  and  the  series  are  both  functions  of  /i,  and  may  be 
differentiated  relatively  to  h. 


DJ(xo-hh)=A,-h2A,h-\-3A,h'-^4A,h'+ -j-nAJi^-'-\- 


We  shall  find  it  convenient  to  adopt  the  following  notation : 
Letf'x  stand  for  D^fx,  f'x  for  DJfx,  /(">a;  for  D^'/x.     Let 


120  DIFFERENTIAL  CALCULUS.  [Art.  124. 

/'^o^/^^^^o  stand  for  the  results  obtained  by  substituting  Xq  for  x 
in  /'x,/("^a:,  where  Xq  may  be  a  single  term  or  any  complicated 
function.     Let  n  !  (which  is  to  be  read  n  admiration)  stand  for 

Ix2x3x4x X  n. 

Call  {.T^-\-Ii)  =  x, 

then  A/(^o  +  li)=DJx  =  DJxD^x 

=f(x,  +  h)  D,{x,  +  h)  =f'{x,  +  h) . 

In  like  manner,  we  could  show  that 

D,V(^o  +  h)=f"{xo  +  h). 

Dh' (^0  +  h)  =/(")  {X,  +  h) ,  &c. 

f'{x,+h)=         A,  -]-2AJi  +  ...--^nAJi--'+...-  ; 
f\x,+h)=  2A,+-^  +  n{n-\)AJi-'+-.-   ; 

Let  /i  =  0  in  these  equations,  and  we  have 

Ao  =  A,  /"'a:o=3!^3, 

/'xo  =  ^„  f^x,=  4.\A,, 

rx,=  2A,,  r-^x,=  n\A^, 

hence  ^o  =  Ao,  ^3  =  ^/'"^o, 

4  ! 


Chap.  IX.]  DEVELOPMENT   IN    SERIES.  121 

and  f{x,  -f-  h)  =Ao  +nf'x,  +  |?^/"xo  +  |^/'"a:o  +  |^/-a^o 

if/(xo  4-^0  ^^^  ^6  developed. 


Examples. 
(1)    To  develop  {a  +  hy. 
Call  (a  +  A)  =  X, 

then  fx  —  x""^ 

fx  —  ?ia;"~\ 

/'"a;=n0^-l)(>^-2)x"-^  &c. 
/a  =  o% 

f"a  =  n{n-l)a''-\ 

f"'a  =  n{n-l){n-2)a^-\  &c. 

(a  H-  A)'*  =  a"  +  na'^-Vi  +  n(n-l)  ^n-2;,2 

^»(n-l)(»-2)^„_3,^3^ ^ 


(/*  (a  +  h)"  can  6e  developed. 

(2)    To  develop  sin/i. 

sin  7i  =  sin  (0  +  h) . 
Let  a;  =  0  +  /i. 


122  DIFFERENTIAL   CALCULUS.  [Art.  125. 

fx=sinx,  yD  =  sinO  =  0, 

/'a7=cosa;,  /'0  =  cosO  =  l, 

f"x  =  -  sina;,  /"O  =  -  sinO  =  0, 

f"'x=-GOSX,  /'"0= -cosO= -1, 

f'''x=s[nx,  &c.  /^^0  =  sinO  =  0,  &c. 

siii(0  +  /AzzzO  +  /i  +  0.—  -  —  +  0.  —  +  —  + , 

^  ^  2!      3!  4!      5! 

sin/i  =  h \- , 

3!      5!      7!      9! 

i/sinh  can  be  developed. 

(3)  Assuming  that  cosh  can  be  developed,  determine  the 
series. 

125.  Let  us  find  what  error  we  are  liable  to  commit  if  we  take 
/(a^o  +  ^O  equal  to  7^  +  1  terms  of  the  series  (Art.  124,  [1]). 
Let  B  be  the  difference  between  /{xq  -{-7i)  and  the  sum  of  the 
first  ?i  + 1  terms  ;  then 

f{x,-i-h)=fx,  +  7tfx,  +  ^f"x,  + +  ^/(")xo+i2, 

2 !  n\ 

and  we  want  to  find  the  value  of  B. 


Lemma. 

126.  If  a  continuous  function  becomes  equal  to  zero  for  two 
different  values  of  the  variable,  there  must  be  some  value  of  the 
variable  between  the  two  for  which  the  derivative  of  the  function 
will  equal  zero. 

For,  in  passing  from  the  first  zero  value  to  the  second,  the 
function  must  first  increase  and  then  decrease  as  the  variable 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  123 

increases,  or  first  decrease  and  then  increase.  If  it  does  the 
first,  the  derivative  must  at  some  point  change  from  a  positive 
to  a  negative  value ;  if  the  second,  the  derivative  must  change 
from  a  negative  to  a  positive  value,  and  in  so  doing  it  must,  in 
either  case,  pass  through  the  value  zero. 

127.   To  determine  R. 

J.n  +  l 

Let  P  =  R-^     '^ 


then  R  =  — P, 

(n  +  1)! 

and 

2 !  w !  (^n-\-\)\ 

or 

f{x,+li)-M-hfx,-^^rx, -i^V)a-o--!^P  =  0. 

2 !  11  \  (n+1)  ! 

Call  (a.'o  +  /0  =  X; 

then  h  =  X—  Xq,  and  we  have 

1 !  nl 

Form  arbitraril}'  the  same  function  of  a  variable  z  that  the  first 
member  of  [1]  is  of  Xq,  and  call  it  Fz. 

Fz=fX-fz-i^^fz-  l^Z^V"^ 

__  (X-zY  .,,^ _  {X-zY^^  p 
n\       *^  (ri  +  1)! 


124  DIFFERENTIAL   CALCULUS.  [Art.   127. 

If  2:  =  a?o,  Fz  becomes  identical  with  the  first  member  of  [1],  and 
therefore  =  0. 

If  z  =  X,Fz=0, 

since  each  term  disappears  from  containing  a  zero  factor ;  and 
we  have  succeeded  in  forming  a  function  of  z^  which  becomes 
equal  to  zero  for  two  values,  Xq  and  X  of  z.  If  Fz  is  continuous, 
there  must  be  some  value  ofz  between  Xq  and  X  for  which  F' z  —  0. 
Differentiating  Fz^  and  remembering  that  P  is  constant,  we  have 


2 


^{X-zl  ^X-zY-^ 

^       2!      -^  ^    ('^-1)  ! 

_ {X~zY  {X-zY  p 

n\       ^  n\ 

All  the  terms  but  the  last  two  destroy  one  another,  and 

F'Z  =  -  (AlZ^7(n+  1)  ^  ^  i^-^Y  p. 

But  this  must  be  equal  to  zero  for  some  value  of  z  between  a^o 
and  X.  Such  a  value  can  be  represented  by  Xq-{-0{X—Xq) 
where  6  is  some  positive  fraction  less  than  1,  i.e.,  0<^<1. 
Substituting  this  value,  we  have 

0  =  -  [^-«^o-^(^-^o)]"j(n+i)  lx^j^O{X  -  a^o)] 
n ! 

[_X-x,-0{X-x,)^^^  p 

Whence  p=/(n  +  i)  [x,+ 0{X-x^)'\. 

X—Xq  =  /i, 


Chap.  IX.]                DEVELOPMENT   IN   SERIES.  125 

whence        f{x,  +  h)  =fx,-\-^^fx,  +  J^^rx,+ 

where  all  that  we  know  about  0  is  that  it  lies  between  0  and  1 . 


128.   The  expression  for  the  last  term  may  be  obtained  in  a 
different  form  by  assuming  at  the  start 

J^n  +  l 

instead  of  E  = — ■  P. 

Making  this  assumption,  show  that 

fi^o  +  h)  =fx,  +  hfx,  +  |^/"a:o  + 

71 !  n  I 

Since  in  each  of  these  formulas  x^^  was  any  given  value,  we  can 
represent  it  in  the  result  just  as  well  by  a;,  and  the  formulas  may 
be  written 

f{x  +  h)  =fx  +hfx  +  |^/"a;  + 

^-J'^^'x  +  jJ^^r-^'^x^oh);  [IJ 


f{x  +  h)  =^fx  +  hfx  +  ^-^J^'x  + 


11  !  n  ! 

and  these  formulas  are  known  as  Taylofs  Theorem. 


126  DIFFERENTIAL  CALCULUS.  [Art.  129. 

Example. 

129.    To  develop  (2+1)^ 

Let  us  see  what  error  we  are  liable  to  if  we  stop  at  the  second 
term. 

fx=x\  f"'x=24.x, 

f'x  =  4:aP,  /^^x=24, 

/"x'=12a.^  f^x  =  0. 

(2 +iy =2'  -{-  iA.2^  -\-  ~  i2{2  -\-  oy. 

If  0  =  0,  the  last  term  is  24.  If  ^  =  1,  the  last  term  is  54. 
Hence,  if  we  stop  at  the  second  term,  our  error  lies  between  24 
and  54.  In  point  of  fact,  it  is  33.  Suppose  we  stop  with  the 
third  term. 

(2  +  l)*=2*  +  1.4.23  +  i!  12.22  + i!  24(2+^). 

If  ^  =  0,  the  last  term  is  8.  If  ^  =  1,  the  last  term  is  12,  and 
the  error  must  be  between  8  and  12.  It  is  actually  9.  Suppose 
we  stop  with  the  fourth  term. 

(2 +1)4  =2^  +  1.4.23+ -11  12. 22 +  i!  24.2  +  — 24. 
Here  the  error  is  precisely  —  24  =  1 . 


^         Example. 
Tofindsin(0+1). 
Let  ?i=7. 

fx=smx,  f^x=:  sin x, 

f'x  =  cosx,  J^x  =  cosx, 

f"x  =  —  sin  a;,  f^^x  =  —  sin  a;,  &c. 
y*^"a;  =  — coscc. 


8! 

=  0. 

8! 

sinl 
40320 

Chap.  IX.]  DEVELOPMENT   IN   SERIES.  127 

sin  (0  H- 1  )=  1  -  —  4-  —  -  —  +  —  sin  ^. 
^  3  !      5  !      7  !      8! 


If  ^  =  0, 


If  ^  =  1, 


tfl^  is  within  ^(j^^^  of  the  true  value  of  sin  1 . 

If  in  any  development  the  general  expression  for  the  eiror 
decreases  indefinitely  as  we  increase  n,  it  follows  that,  as  the 
number  of  terms  of  the  series  is  indefinitely  increased,  the  sum 
will  approach  as  its  limit  the  value  of  the  function,  which  is 
therefore  equal  to  a  series  of  the  form  obtained,  and  is  said  to  be 
developable. 

130.    Let  us  consider  some  examples. 
To  develop  log(l  +  a:) . 
Let2:=(l-f-a;). 

/2;  =  log^, 

/(n)^^(_l)n-l(^_l)t^-n^ 

/-(I)  =2, 


128  DIFFERENTIAL  CALCULUS.  [Art.  130. 

/-(l)=-3!,^ 

/(">(1)  =  (-1)"-Hn-1)!, 

/{«  +  i)(l_|_^a;)  =  (_l)«yi!(l+^a;)-"-i. 

By  Taylor's  Theorem, 

log(l4-a;)  =  a; --  +  ---  + +  (_l)~-i^ 

?i  + 1 
The  ratio  of  the  nth  term  to  the  term  before  it  is , 

or  —  (  1  —  \x.    If  ic  is  greater  than  1  in  absolute  value,  f  1  —  ]x 

\       n)  \       n) 

will  eventually  become  and  remain  greater  than  unity  as  n  in- 
creases, and  the  series 

01?   ,    'X?       x^  - 

X f- 

2       3       4 

is  divergent  and  cannot  be  equal  to  log(l-}-.T).  So  we  need 
only  investigate  the  expression  for  the  error  for  the  values  of  x 
between  +1  and  —1.     Suppose  x  is  positive,  and  less  than  1. 

Then  (1-j- ^^x)~"~^  approaches  zero  as  its  limit  as  n  in- 

creases    indefinitely,    for    it    may   be    thrown    into    the    form 

f--^— )      .      Since  a;  <1,  -— ^— <  ^      ( — ^V  +  i   i^^s 

zero  for  its  limit  as  n  increases  indefinitely ;    as  has  also  the 

factor .    Hence,  for  values  of  x  between  0  and  1,  log(l  +  x) 

is  developable,  and  is  equal  to  the  series 

X'    ,    X?        iC^    , 

X V- 

2       3       4 

This  is  true  even  where  ic  =  1 ,  for  it  is  easily  seen  that,  in  that 

1      /     X    \"  +  i 

case  also, ( )        approaches  the  limit  zero  as  n  in- 

n-\-\  \l-\-  Ox  J 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  129 

creases.  If  x  is  between  0  and  —1,  the  second  form  of  the 
error,  Art.  128,  [2],  is  most  convenient  for  our  purpose.  Let 
x=  —x\  so  that  x'  is  positive  and  less  than  1 .  Then  our  func- 
tion is  log(l  —  ic'),  and  the  series  becomes 

2        3        ~~n~    {l-Ox'y  +  ^ 


i-ox'y+^  ~  i-ox\i-ox') 


x' f)x' 

where ■  is  less  than  1  ; 

1-Ox' 

hence  limit /.V-^V^ 

-^--\l-(jx'J 


71=  X 


and  as j  is  a  finite  value,  the  expression  for  the  error  de- 
creases indefinitel}'  as  n  increases,  and  the  function  is  equal  to 
the  series.     Our  expansion 

/>»*  /mo  /y»4 

log(l  +  x')  =  a;-|-h|-|-f- 

holds,  then,  for  values  of  a;  between  1  and  —1. 

The  Binomial  Theorem. 

131.   To  develop  (1  +  .^)"*. 

Let  j2  =  l  -\-x, 

f2  =  ^"', 

f'z=m2^~\ 

f"2  =  m{m—l)^-\ 

f'"0  =  m(m  -1)  (m  -  2)^-^ 

/(">^=m(m  — 1) (??i  — n  +  l)^'"-", 

/<"  +  >>^  =  m(m  — 1) {m  —  n)z"'-''-\ 


130  DIFFERENTIAL   CALCULUS,  [Art.  13L 

/'(l)=m. 

/"(l)  =  m(m-l), 

/"'(l)=m(m-l)(m-2), 

/W(l)  =  m(m-1) (m-n  +    ), 

/(«  +  !)  (1  4.  ^a:)  =  m(m  -1) (m  -?i)  (1  +  ^/a;)'"-"-^ 

By  Taylor's  Theorem, 

/I   I     \m     1  ,  .  m  (m  —  l)x^      m(m  —  1)  (m  —  2)x'^  . 

2!  3! 

If  m  is  a  positive  whole  number, 

/('">^  =  m  !, 

and  all  succeeding  derivatives  are  0,  so  in  that  case  (l  +  a?)"*  is 
equal  to  the  sum  of  a  finite  number  of  terms,  namely  (m  +  1) 
terms.  If  m  is  negative  or  fractional,  however,  this  is  not  the 
case.  Let  us  see  whether  (1  H-x)"*  is  then  developable.  The 
ratio  of  the  general  term  of  the  series  to  the  one  before  it  is 


Ijx.     If  aj  is  numerically  greater  than  1 


n  \    n 

this  ratio  will  eventually  become  and  remain  greater  than  1  in 
absolute  value  as  n  increases,  and  the  series  is  divergent  and 
cannot  be  equal  to  the  function.  Hence  we  need  examine  the 
value  of  the  error  only  for  values  of  x  between  1  and  —  1 .  The 
expression  for  the  remainder  after  n-\-l  terms  is 

which  may  be  thrown  into  the  form 

rm(m-l) (m-n)  ^^^,1  1 

L  (n  +  1)!  j{l+Oxy+^-"'' 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  131 


As  n  increases,  the  limit  approached  by  x^  +  i-m  ^®   "^* 


(1  +  oxy 

greater  than  1 .     Increasing  n  b}'  unity  multiplies  the  quantity 
in  parenthesis  by  —^ — '^^  x^  which  may  be  written 


fm-1 ?i_\ 

\n-\-2      n-\-2j 


and  by  taking  n  sufficiently  large,  this  multiplier  may  be  brought 
as  near  as  we  please  to  the  value  —x.  If  a;  lies  between  0  and  1 , 
—  X  is  numericall}'  less  than  1  ;  and  as  n  increases  indefinitel}',  we 
multiply  our  parenthesis  b}-  an  indefinite  number  of  factors,  each 
less  than  1 ,  and  so  decrease  the  product  indefinitely.  Therefore, 
for  values  of  x  between  0  and  1 ,  the  expression  for  the  error 
approaches  zero  as  its  limit  as  n  increases  indefinitely,  and 
(1-f-  a;)*"  is  equal  to  the  series 

l  +  ma^  +  ^^^-^>  ^_^rn(m-l){m-2)  ^     ^  ^ 


Example. 


Show,  by  considering  the  second  form  for  the  error,  Art.  128, 
f2],  that  for  values  of  x  between  0  and  —1,  (l  +  ic)"*  is  devel- 
opable. 

The  Binomial  Theorem  follows  easil}'  from  the  development 
of  (l  +  x)«. 

and  if  7i  is  less  than  x  in  absolute  value,  we  have 

(x+h)^=x'^-^mx^-''h -\-'^^'^~^'^  x^-Vi^ 

+  m(m-lKm-2)  ^,.3^3 _^ ^  ^2] 

no  matter  what  the  value  of  m. 


132  DIFFERENTIAL  CALCULUS.  [Art.  132. 

Of  course,  if  h  is  greater  than  x,  we  can  write  (a?  -f  /i)"*  in  the 
form  /i"*!  1  -f-  -  J  ,  and  shall  then  get  as  a  true  development, 


(/i  Jf-x)'^  =  /r  +  mle^-^x  + 


Maclauriyi's  Theoreyn. 
132.    If,  in  Art.  128,  [1]  and  [2],  we  let  x  =  0,  we  get 

fh=f{^)  +  /^/'(O)  + 1^/"(0)  +  |^/"'(0)  + 

fh  =/(o) + r  (0) + |^/"(o) + |^/'"(o) + 

n  !  ^  ! 

It  does  not  matter  what  letter  we  use  for  the  variable  in  these 
formulas.     Change  li  to  a.',  and 

/x=/(0)  +  x/'(0)  +  ^/"(0)  + +  ^/<"'(0) 

(«+])r  ■- J 

>  =/(o) + »/'(o) + ^/"(O)  + + ^/'"'(O)     ■ 

71  ! 

These  results  are  called  Madaurin's  Theorem^  and  they  enable 
us  to  develop  a  function  in  a  series  arranged  according  to  the 
ascending  powers  of  the  variable. 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  133 

133.    To  develop  a\ 

>  =  a%  /(0)  =  a«  =  l. 

f'x  =  anoga,  /'(O)  =  loga, 

f"x  =  a^(log«)S  /"(O)  =  (log«y, 

/(")a;  =  a^(loga)%  /^"K0)  =  (loga)", 

/(«+i)a;  =  a^(loga)"  +  \  /('*  +  i^^a;  =  a^*(loga)"+^ 
By  Art.  132,  [1], 

a^  =  l  +  xlog«  +  ^(loga)2H-|^(loga)3  + H--,(loga)" 

iC''+^(loga)"+^    Q^_    Q^  a^loga  x\oga  a^loga       a^loga  a;loga 
____       a    -a    .—^  -  ^ __.___. 

No  matter  what  v^lue  x  may  have,  after  n  has  attained  a  cer- 
tain value  in  its  increase,  some  of  the  factors  of  this  product 
will  approach  the  limit  zero,  and  the  whole  product  will  there- 
fore have  zero  for  its  limit  as  n  increases  indefinitel}',  and 

a^  =  l  +  xloga  +  |^(loga)2+^(logar+ [1] 

for  all  values  of  a;.     If   a  =  e,  log  a  =  1 , 

and  e^l^--^—^-  —  -^  —  -^ [2] 

12  !      3!      4!  •-  -^ 

Let  a;  =  1 ,  and  [2]  becomes 

e  =  H-i-hi-  +  -i-  +  J-  + ;  [3] 

12!      3!4! 

a  result  already  established  in  Arts.  61  and  62. 


134  DIFFERENTIAL   CALCULUS.  [Art.  134. 

134.  We  can  now  test  the  accuracy  of  the  provisional  devel- 
opments of  sine  and  cosine  given  in  Art.  124,  (2)  and  (3).  By 
Art.  132,  [1], 

smx  =  x-  —  +  ~-~-u 4-i?, 

3!      5!      7!  ' 

where  Ii=  ^^^ll^jxn+i^ ox=  ±-^^^  sin^x 


or  ±- —  cos/^ic. 

(n  +  l)! 

In  either  case,  one  factor  sin^a?  or  cosmic  is  between  1  and  —  1, 
and  the  other  approaches  zero  as  ?i  increases  indefinitely  ;  there- 
fore, sina7  =  a;  — —  H--^: — fL-L.^  — 

3!      5!      7!       9! 


Prove  that   cosa;  =  l  — —  4-^  — £!  j-^  _ 

2  14-!       fi  I       «  t 


Example. 

2"!      r!~6!  '  8 


135.    By  the  aid  of  the  Binomial  Theorem,  tan~^a;  and  sin-^a; 
can  be  very  easily  developed. 

DMn-'x=-^  =  (l-\-x')-\   (Art.  71,  Ex.) 

For  values  of  x  less  than  1,  (l  +  xF)-'^  can  be  developed  by  Art. 
131,  [2],        {l-^x')-^  =  l-a^  +  x'-x'  +  x'-..-. 
Integrate  both  members. 

ian-'x  =  C+x-t-^^-^  +  ^- 

3^5       7^9 

To  determine  our  arbitrary  constant  O,  let 

x=0; 


Chap.  IX.]  DEVELOPMENT  IN  SERIES.  135 

then  tan-iO  =  C+0-5 , 

o 

and  C=0. 

/y^  /yi5  /-w»7  /^y 

tan-^x  =  x— -  +  -  —  —+- — [1] 

o  0  i  J 

when  ic  is  less  than  1  ;  that  is,  when  tan~^x  is  less  than  -. 

4 

Asin->^=— J— -=(l-x')-5,      by  Art.  71. 
^/^l  — iC  ) 

For  values  of  a;  less  than  1,  (1  —  a^)-i  can  be  developed  by  Art. 
131,  [2]. 

(l-^)-.  =  l  +  l»^  +  M.^+lJ|^  +  |fM^  + 

Integrating 

.1        ^  ,       ,  1  a.-^  ,  1.3  a^  ,  1.3.5  a?^, 
s,n-,  =  C  +  .  +  --  +  — -  +  — ^-  + 


When  x=0, 

sin~^x=0  and  (7=0. 


.1  ,  1  a^  ,  1.3  ar^  .  1.3.5  a;^^ 


Examples. 

(1)    Show  that  sin(x  +  h)  is  equal  to  the  series 

h  h^  /i^  h*    . 

sin  a;  +  :p  cos  a;  -  —  sin  a;  -  —  cos  a;  +  —  sina;  + 


(2)    Show  that 


^^   ■  -I  ,  ^a;      m^a^^      m^a;^ 
~    "^  1!^    2!    ^    3! 


136  DIFFERENTIAL  CALCULUS.  [Art.  136. 

136.  Although  the  strict  proof  that  any  given  function  is  equal 
to  the  series  obtained  by  Ta3'lor's  Theorem  requires  the  investi- 
gation of  the  remainder  after  n  +  1  terms,  it  is  often  convenient 
to  obtain  terms  of  the  series  in  cases  where  the  expression  for 
the  remainder  is  too  complicated  to  admit  of  the  usual  examina- 
tion. When  such  a  series  is  employed,  it  is  to  be  remembered 
that  it  is  equal  to  the  function  in  question  onl}^  provided  that 
the  function  is  developable.  Sometimes  the  possibilit}'  of  de- 
velopment can  be  established  by  other  considerations,  and  some- 
times in  rough  work  no  attempt  is  made  to  fill  out  the  proof  of 
the  assumed  equality. 

Examples. 

(1)  Develop  -^  +  log(l  +  aj). 

l-\-x 

Ans.    2x--ix^  +  -a^--x*-{--sc^  + 

2  3  4  5 

(2)  Obtain  4  terms  of  the  development  of  log(l  +  e*) . 

Ans,   log2+-  +  ^        ^ 


2      2^      2^.4  ! 

137.  In  the  work  of  successive  differentiation  required  in 
appl3'ing  Taylor's  Theorem,  a  good  deal  of  labor  can  often  be 
saved  by  making  use  of  Leibnitz's  Theorem  for  the  Derivatives 
of  a  Product.      Let  y  and  z  be  functions  of  x.      Represent 

D,y,  D^'y, Dj^y  by  y',  y", 2/(">  and  D,z,  D^'z, Z>/^  by 

z',  z", ^w. 

D^\yz)  =  y"z  ^2y'z' -{-yz", 

i>.\y^)  =  y'"^  +  sy  2'+  syz"  +  yz"\ 

DJ(yz)  =  y'^z  +  4y"'z'-h  6y"z"+4:y'z"'-{-yz'^. 

Examining  these  results,  we  see  that  the  coefficients  of  the  terms 
in  the  successive  derivatives  are  the  same  as  in  the  correspond- 
ing powers  of  a  binomial,  and  that  the  accents  follow  the  same 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  137 

law  as  the  exponents  in  the  powers  of  a  binomial.  Following 
the  same  analogy,  we  should  have 

n{n-\){7i-2)     (n-3)^'/'  , 
3!  ^ 

Assuming  for  the  moment  the  truth  of  this  equation,  let  us  dif- 
ferentiate both  members.     We  obtain 

(yi  +  l)n(yi-l)  ,^^,„       .   . 

3! 

but  this  is  precisely  what  we  should  expect  for  the  {n  +l)st  de- 
rivative from  the  observed  analog}'.  Hence,  if  our  rule  holds  for 
the  ?ith  derivative,  it  holds  for  the  (?i  -j-l)st ;  but  we  have  seen 
that  it  holds  for  the  4th,  therefore  it  holds  for  the  5th,  and 
therefore  for  the  6th,  and  so  on ;  and  it  is  in  consequence 
universally  true.  This  rule  is  called  Leibnitz's  Theorem^  and  is 
formulated  as  follows : 

_l_  n{n-\)(n-2)  y„_3)^m |-j-| 

138.   Assuming  that  tana;  can  be  developed,  let  us  obtain  a 
few  terms  of  the  series.     Here 

fx  =  tan  x  =  y, 

f'x  =  2/'  =  sec^ic, 

f"x  =  y"  =  2  8e(^xtanx=  2  y'y, 

rx  =  y"'=2(y'fy+y'y'), 

f^-x  =  y^=  2{y'"y  +  2y"y'+  y'y") , 


138  DIFFERENTIAL   CALCULUS.  [Art.   139. 

rx=y''  =2(7/-7/+3y"2/'+3yY'+7/'y"), 

&c.,  by  Leibnitz's  Theorem. 


When 

x  = 

=  0, 

^  =  0, 

2/"  =  0, 

y=l, 

2/^=10, 

2/"=0, 

2/-  =  0, 

2/'"=  2, 

2/^"  =  272. 

By  Maclaurin's  Theorem 

+  ,    2     o  ,  16    .  ,  272    7  , 

3 !  5  !  7! 


Example. 
Assuming  that  secic  can  be  developed,  show  that 

secaj  =  1 H 1 h 


2!       4!         6! 


Indeterminate  Forms. 


139.  The  subject  of  indeterminate  forms  is  readily  dealt  with 
by  the  aid  of  Taylor's  Theorem.  Take  the  form  -.  Suppose 
fx  and  Fx  are  functions  of  a;,  continuous  for  values  of  x  near  the 

particular  value  a,  and  fa  and  Fa  are  both  equal  to  zero,  to  find 

fx 
the  true  value  (vide  Art.  34)  of  ^^^  when  x  =  a. 
^  ^       Fx 

Call  a;  — a  = /i,  then  «  =  a  +  /i, 

and  we  can  develop /x  and  Fx  by  Taylor's  Theorem. 


Chap.  IX.]  DEVELOPMENT   IN    SEKIES.  139 

fx  =f{a  +  h)  =fa  +  hf  (a  +  Oh) 
where  6  is  some  number  between  zero  and  1 . 

Fx  =  F{a  +  h)  =  Fa-\-  hF'{a  +  d'h) 

where  0<^/<l. 

fx^  _  hf\a  +  Oh)   _  f'(a-\-Oh) 
Fi  ~  hF'{a  +  O'h)  ~  F'ia  +  ^70 ' 

since  /«  =  0  and  Fa=0. 

As  a?  approaches  a,  h  approaches  zero  ;  hence  Oh  and  o'h  approach 

fx 
zero  as  their  limit ;  consequently  the  limit  approached  by  -—  as 

J1 X 

fa 
X  approaches  a,  is  •— — ,  which,  by  Art.  34,  is  the  true  value  of 
F  G> 

^.     If       /a  =  0,  Fa  =  0,  f'a  =  0,  and  F'a  =  Oy 
Fa 

it  will  be  necessary  to  carry  the  development  one  step  farther. 
fx  =f(a  +  h)  =fa  +  hfa  -f-  ^^/"(a  +  Oh)  =  f/'{a  +  Oh) , 

Fx  =  F{a  +  h)  =  Fa  +  hF'a  +  —  F"{a  +  O'h)  =  ^  F" {a  +  Oli), 

and  ^_f"{a  +  Oh) 

Fx~  F"{a+oniy 

f"a 
which  approaches  ^  ^^     as  its  limit  as  x  approaches  a. 

Example. 

Show  that,  if /a,  i<^a,  /'a,  F'a,  f"a,  F"a,  &c.,  /('*-^>a,  and 
2^(n-i)^  all  equal  zero,  the  true  value  of  •^—  when  a;  =  a  is  i^— - — 

140.    The  reasoning  of  the  last  section  does  not  apply  when 


140  DIFFERENTIAL   CALCULUS.  [Art.  140. 

a  =  00  ,    as   then  f{a  -j-  h)    cannot   be   developed    by   Taylor's 
Theorem. 

fx 
To  find  the  true  value  of  '^—  when  a;=  oo,  supposing  that 

fx=0  and  Fx  =  0  when  x=  cc. 
Let  2/  =  -, 

X 

then  fx  =/-  and  Fx  =  F-,  and 

^  2/  2/ 

•^- 

y  0 

—  assumes  the  form  -  when  y  =  0, 

j,l  0  ^       ' 

and  its  true  value  for  w  =  0  will  be — 

L      yjy=o 
y       y     y       y     y 

DyFl  =  -\Fl. 

y       y     y 

2/ 
But  the  value  of when  2/  =  ^ 

DyF- 

y 

is  the  value  it  approaches  as  y  approaches  0. 
D.f-      --J'-      /'' 


y        y^    y       y 

D,F-      --F'-     F^- 

y       y     y       y 

~F'x' 

but  when                            2/  =  0,  a?  =  oo  ; 

fx 
hence  the  true  value  of  •—  when  a;  =  oo 
Fx 

fx 
is  the  value  of  *4--  when         x  —  co\ 

F'x 


Chap.  IX.]  DEVELOPMENT  IN   SERIES.  141 

and  the  method  of  the  last  section  holds,  no  matter  what  the 
value  of  a. 


141.    It  was  shown  in  Art.  35,  that  the  form  ^  could  always 

be  reduced  to  -  and  treated  as  above.    Let  us  consider  a  general 

example.     Suppose   /a  =  qo  and  i^a  =  qo  , 

fx 
required  the  true  value  of  ^—  when  x=a. 


1 

>=^  =  5when.= 
Fx        1        0 

a. 

/^ 

Differentiate  numerator  and  denominator. 

^>=-(yW^^- 

^-A=-(Flr^'- 

Hence 

we  have,  when             x=a^ 

A 

^      F'x 

F'x 

Fx 

-     1  ^,,  W 

^^fxF'x 
Fx  f'x 

f'x' 

and 

fx      f'x 
Fx     F'x 

the  value  required.    Therefore  the  form  ^  can  be  treated  directly 
by  the  same  method  as  the  form  -.     In  dividing  both  members 


142  DIFFERENTIAL  CALCULUS.  [Art.  141. 

Of  the  equation  ^^  =  ^^Y  —  by  '^, 

^  Fx      \FxJ  f'x  ^  Fx 

fx 
we  have  assumed  that  the  true  value  oi-^—^  when 

Fx 

x=  a 

is  neither  0  nor  go  .      Suppose  the  limit  approached  by  ^  as 

X  approaches  a  is  0,  and  that  fx  and  Fx  increase  indefinitely. 

Form  the  function  f^-^^^, 
Fx 

Its  true  value  when  x  =  a,  is,  of  course,  1  ; 

but  when  x  =  a,  it  assumes  the  form  —  : 

hence  its  true  value  when       a;  =  a, 

f'x  4-  F'x 

must  be  the  limit  approached  by*^ — ~ as  x  approaches  a, 

Jo  X 

which  is        i+'™'*r^i- 

TJ,nr.oiv.v^  limit  r/'^l     ^_ limit  r/«1       ,     ,        ^,     . 

Therefore,  ^^^  |^^,^J  =  0  -^^^  ^^J       by  hj^othesis. 

fx 
If  the  true  value  of  •^—  when   x  =  a 
Fx 

is  infinite,  of  course  the  true  value  of  its  reciprocal  —  will  be 
zero,  and  will  equal 


hence  T^         =  '^ 


iF'xi^^ 


Fx 


and  the  method  of  determining  the  form  — ,  established  at  the 


GO' 


beginning  of  this  section,  is  of  universal  application. 


Chap.  IX.]  DEVELOPMENT   IN  SERIES.  143 

142.  The  forms  oc°,  1°°,  0°,  can  all  be  reduced  to  one  of  the 
forms  already  discussed,  if  we  make  use  of  logarithms.  It  is  to 
be  observed^  that  these  forms  to  be  indeterminate  must  all  occur 
as  limiting  fonns  of  a  function  of  tivo  functions;  and,  in  order 
that  the  forms  may  admit  of  being  determined,  the  two  functions 
must  depend  upon  the  same  variable. 

Let  u  =  (Fx)^\ 

Suppose,  when  x  =  a., 

Fx=cc  and  fx=0  ; 

to  find  the  true  value  of  u  when  x  =  a. 

log  u  =fx .  log  Fx  =  0  X  oo  when  x  =  a, 

and  may  be  determined  by  the  method  of  Art.  35. 

Examples. 

(1)  Show  that  1*,  0°,  can  be  made  to  depend  upon  the  forms 
00  X  0  and  0(  — oc). 

(2)  Obtain  a  method  for  dealing  with  the  form  oo  —  oo. 
Find  the  true  value  of  the  following  functions  ;  — 

(3)  i^i^  when  x  =  l.  Ans.    1. 

X  —  A 

(4)  ^IzifT  ''      x  =  0.  Ans.    2. 

sinic 

(5)  ^-sin-^a;  ,,      ^^^  ^^^     _^ 

sin^a; 

(6)  ~ ^  ''      x=l.  Ans,    -1. 

log  a;      logo; 

(7)  ^'-2cosa:  +  e-  ,,      ^^^^  ^^^^^    2. 

a;  sin  a; 


144  DIFFERENTIAL   CALCULUS. 

(8)  xtsLiix seca;  when  x  =  -. 

(9)  2=^sm^  "      a;=:cx.. 
(10)  (al-l)a;  "      x  =  cc. 

(11)  r^+iY         "  ^=^. 

(12)  (l  +  iy  ''      0^  =  00. 

(13)  A^"^Y  ''      a;  =  0. 

(14)  /^^^"^y^  "      x  =  0. 


(15) 


/tanajV 


[Art. 

143. 

Ans.    - 

-1. 

Ans. 

a. 

Ans.    log  a. 

Ans. 

e«. 

Ans. 

1. 

Ans. 

1. 

Ans. 

eK 

^  "      x  =  0.  ^ns. 


(16)    s'mx''''''  ''      x  =  -.  Ans.    1. 

^     ^  2 


Maxima  and  Minima. 

143.  Taj'lor's  Theorem  enables  us  to  give  a  very  simple  and 
complete  treatment  of  the  subject  of  maxima  and  minima  of  a 
a  single  variable. 

Let  fx  be  a  function  of  a;,  finite  and  continuous  for  values  of 
X  near  the  particular  value  a. 

Call  x  =  a-\-h. 

/{a  +  h)  =fa  4-  hf'a  +  ^f'(a  +  0h) . 

/(a  +  h)  -fa  =  /(/-'a  +  |^  /"(a  +  ^/i). 


Chap.  IX.]  DEVELOPMENT   IN   SERIES.  145 

In  order  that  fa  should  be  either  a  maximum  or  a  minimum, 

f{a^li)-fa 

must  have  the  same  sign  for  small  values  of  h  whether  li  is  posi- 
tive or  negative.  If  this  sign  is  minus,  fa  is  a  maximum  value 
of /«  ;  if  plus,  a  minimum  value  {vide  Art.  39). 

If /'a  does  not  equal  zero,  we  can  take  a  value  of  U  so  small, 
that,  for  it  and  all  smaller  values, 

shall  be  less  than/'«.     The  sign  of 

will  then,  as  li  approaches  zero,  ultimately  become  and  remain 
the  same  as  the  sign  of  Uf^a  ;  but  the  sign  of  Ufa  changes  with  the 
sign  of  /i,  so  that  fa  can  be  neither  a  maximum  nor  a  minimum. 


144.    Suppose  fa  =  0, 

then  /(a  +  h)  -fa  =  ^  fa  + 1^  /'"(a  +  Oh) 


li" 


2!       3 ! 


as  li  approaches  zero       — f"'(a-\-  Oh) 
o  ! 

f"a 
will,  in  the  end,  become  and  remain  less  than-^^^ —  and  the  quan- 

tit}"  in  parenthesis  will   have  the  same  sign  as  fa.     As  h^  is 
necessarily  positive  for  all  values  of  h 

f(a-\-h)-fa 

will  then  be  negative  for  small  positive  and  negative  values  of  h, 


146 


DIFFERENTIAL   CALCULUS. 


[Art.  145. 


if /"a  is  negative,  and  will  be  a  maximum;  if /*  a  is  positive, 
fa  will  be  a  minimum. 

145.  It  can  be  easily  established  by  an  extension  of  the  rea- 
soning of  the  last  section,  that,  if  the  first  derivative  that  does 
not  vanish  when  x  =  a,  is  of  odd  ordeT^^  fa  is  neither  a  maximum 
nor  a  minimum;  that,  if  it  is  of  even  order  and  negative,  fa  is  a 
maximum;  if  of  even  order  and  positive,  fa  is  a  minimum. 

Examples. 

(1)  A  body  moves  with  different  uniform  velocities  in  two 
different  media  separated  by  a  plane,  required  the  path  of  quick- 
est passage  from  a  given  point  in  the  first  medium  to  a  given 
point  of  the  second.  It  is  easily  seen  that  the  required  path 
will  he  in  a  plane  passing  through  the  two  given  points  and 
perpendicular  to  the  plane  separating  the  two  media. 


D          C 

9>^ 

I 
p 

Q 

E 

8 

Let  ACB  represent  any  such  path  from  A  to  B.     Draw  a 
normal  to  the  plane  at  C  and  the  perpendiculars  p  and  q.     Call 


and  let  Vi  and  V2  be  the  velocities  in  the  first  and  second  media 
respectively. 

J.(7  =  psec^, 

C^=i)tan^, 


Chap.  IX.]  DEVELOPMENT  IN   SERIES.  147 

BC=  qsecOi, 

DC=qtanOi, 

ptsLYiO  -^  qtanOi  =  c. 

AC  _  J)  sec  0 

is  the  time  required  to  pass  from  ^  to  C ; 
BC  _qsecOi 

is  the  time  required  to  pass  from  O  to  jB  ; 

.     pseeO      qsecOi 
''  — 1 7. — 


is  the  function  we  wish  to  make  a  minimum.     0  and  0^  are  the 
only  variables  in  ?,  and  they  are  connected  b}^  the  relation 

j9tan^  +  gtan^i=  c. 

P  Q 

Dot  =  —  sec  0  tan  0  -\ —  sec  0^  tan  Oi  Do  d-^. 

Differentiate  p  tan  0-\-q  tan  ('i  =  c. 

p  sec^  0-{-q  sec^  O^DqO^  =  Q, 

q  sec-  Oi 

p  q  psec?0 

Dnt  =  —  sec  ^  tan  0 sec  <?itan  0-, 2^» 

(^       Vi  V2  q  seer  0^ 

Dgt  must  equal  zero  in  order  that  t  may  be  a  minimum.     Ex- 
press everything  in  terms  of  sine  and  cosine. 

p  sin  0       q    sin  0i  p  cos^  ^1  _  n 
Vi  cos^  0      V2  cos^  Oi  q  cos^  0 


148  DIFFERENTIAL   CALCULUS.  [Art.  145. 

sin  0      sin  ^i 


Vl  V2 


sin^ 


sin  (>i      V2 
By  taking  D^t  and  substituting 

sin  0  _  Vi 

sin  0^      V2 

we  should  obtain  a  positive  result ;  so  that  this  relation  between 
the  angles  gives  the  path  of  quickest  passage  required.  This 
result  is  the  well-known  law  of  the  refraction  of  light,  and  our 
solution  establishes  the  fact  that  a  ra}'  of  light,  in  passing  from 
a  point  in  one  medium  to  a  point  in  another,  takes  the  course 
that  enables  it  to  accomplish  its  journey  in  the  least  possible 
time. 

(2)  What  value  of  x  will  make  sin'' x cos x  a  maximum? 

Ans.    x  =  -. 

a 

(3)  What  value  of  x  will  make  sina;(l4-  cosx)  a  maximum? 

Ans.   x  =  -' 
3 

(4)  Show  that  x^  is  a  maximum  when  x  =  e. 

(5)  A  statue  a  feet  high  stands  on  a  column  b  feet  high  ;  how 
far  from  the  foot  of  the  column  must  an  observer  stand  that  the 
statue  may  subtend  the  greatest  possible  visual  angle  ? 

Ans.    V&(a-f-6)  feet. 

(6)  Required  the  shortest  distance  from  the  point  (xQ^yo)  to 

the  line  Ax-\-By+C=0. 

Ans.    -^^o  +  ^gyo+C; 


Chap.  X.]  INFlIsriTESIMALS.  149 


CHAPTER    X. 

INFINITESIMALS. 

146.  An  infinitesimal  or  infinitely  small  quantity  is  a  variable 
which  is  supposed  to  decrease  indefiriitely ;  in  other  words,  it  is  a 
variable  which  approaches  the  limit  zero. 

What  we  have  called  the  increment  of  a  variable  has,  in  every 
case  considered,  been  such  a  quantity ;  and  what  we  have  called 
a  derivative  has  been  the  limit  of  the  ratio  of  infinitesimal  incre- 
ments of  function  and  variable. 

147.  When  we  have  occasion  to  consider  several  infinitesimals 
connected  b}'  some  law,  we  choose  arbitrarily  some  one  as  the 
principal  infinitesimal. 

Any  infinitesimal  such  that  the  limit  of  its  ratio  to  the  princi- 
pal infinitesimal  is  finite,  is  called  an  infinitesimal  of  the  first 
order. 

An  infinitesimal  such  that  the  limit  of  its  ratio  to  the  square 
of  the  principal  infinitesimal  is  finite,  is  called  an  infinitesimal 
of  the  second  order. 

An  infinitesimal  such  that  the  limit  of  its  ratio  to  the  nth  power 
of  the  principal  infinitesimal  is  finite,  is  called  an  infinitesimal 
of  the  nth  order. 

Let  «  represent  the  principal  infinitesimal,  and  «i  any  infini- 
tesimal of  the  first  order,  o.^  of  the  second  order,  «„  of  the  ?ith 
order.     Then,  by  our  definition. 


limit  —  =  K^ 
^  being  a  finite  quantity. 


150  DIFFERENTIAL   CALCULUS.  [Art.  148. 

where  e  is  an  infinitesimal  (Art.  7) , 

limit -^=^'; 

On 

Examples. 

Show,  by  the  aid  of  these  expressions,  that  the  limit  of  the 
ratio  of  any  infinitesimal  to  one  of  the  same  order  is  finite  ;  to 
one  of  a  lower  order  is  zero  ;  to  one  of  a  higher  order  is  infinite. 
That  the  order  of  the  product  of  infinitesimals  is  the  sum  of  the 
orders  of  the  factors,  and  that  the  order  of  the  quotient  of  infini- 
tesimals may  be  obtained  by  subtracting  the  order  of  the  denomi- 
nator from  the  order  of  the  numerator. 

Show  that,  if  the  limit  of  the  ratio  of  two  infinitesimals  is 
unit}^  they  differ  by  an  infinitesimal  of  an  order  higher  than 
their  own. 

148.  The  sine  of  an  infinitesimal  angle  is  infinitesimal ;  for, 
as  the  angle  approaches  zero,  the  sine  approaches  zero  as  its 
limit. 

If  we  take  the  angle  as  our  principal  infinitesimal,  the  sine  is 
an  infinitesimal  of  the  first  order  ;  for  we  have  seen  that 


limit 
a=0 


^]-' 


(Art.  68). 


The  vers  a  is  infinitesimal  if  a  is  infinitesimal,  for 
vers  a  =  1  —  cos  a  ; 


Chap.  X.] 

INFINITESIMALS. 

and  as 

a=  0,  COS  a  ==1  ; 

hence 

versa  =  0. 

151 


It  is  an  infinitesimal  of  a  higher  order  than  the  first,  for  we  have 

seen  that  ""o  [^'^^J  =  <>'  (Art.  68). 

Let  us  see  if  it  is  of  the  second  order ;  that  is,  let  us  see  if 
limit  p-cosa'j  .    ^  .  1-cQsa  the  form  -  when 

«  =  0,  and  we  can  find  our  required  limit  by  the  method  of  Art. 

139,  which  gives  us  -  as  the  value  sought.    Therefore,  when  a  is 

id 
infinitesimal,  versa  is  infinitesimal  of  the  second  order. 


Examples. 

Taking  a  as  the  principal  infinitesimal,  show  that 

(1)  tana  is  an  infinitesimal  of  the  first  order. 

(2)  a  —  sin  a  is  an  infinitesimal  of  the  third  order. 

(3)  tan  a  —  a  is  of  the  third  order. 

149.  Let  y  be  an}-  function  whatever  of  a;,  if  we  give  x  an 
infinitesimal  increment  Jx,  the  corresponding  increment  Jy  of  y 
will  be  an  infinitesimal  of  the  same  order  as  Jx,  unless  for  par- 
ticular single  values  ofx. 

To  establish  this  proposition,  we  must  show  that   }  _^^    -r-, 
is  finite,     j  ,_i.q    j-     cannot  be  zero,  except  for  single  values 

of  a; ;  for,  suppose  it  could  become  and  continue  zero  ;  j^^  a    T" 

is  Z>^2/,  and  we  have  seen  (Art.  38)  that  D^y  shows  the  rate  at 
which  ?/  is  changing  as  x  changes.  If  D^y  becomes  and  remains 
zejK),  y  does  not  change  at  all  as  x  changes ;  and,  therefore,  is 
not  a  function  of  x,  but  a  constant. 


152 


DIFFERENTIAL   CALCULUS. 


[Art.  150. 


limit 
Ax=0 


Ay 
Ax 


case. 


limit 


cannot  become  and  continue  infinite  ;  for,  in  that 

Ax~ 


_Ay_ 


would  be  zero,  D^x  would  be  zero,  and  x, 


regarded  as  a  function  of?/,  would  be  constant. 


Since 


limit 
Ax=0 


Ax 


can  be  neither  zero  nor  infinite,  it  must  be 


finite,  and  Ay  and  Ax  are  of  the  same  order. 


150.  If  the  coordinates  of  the  points  of  a  curve  are  expressed 
as  functions  of  a  third  variable  a,  the  distance  between  ttvo  infi- 
nitely near  points  of  the  curve  is  an  infinitesimal  of  the  same  order 
as  the  difference  between  the  values  of  a  to  ivhich  the  points  corre- 
spond. 

The  ordinary  equations  of  the  c3'cloid, 

X  =  aO  —  a  sin  0 


yz=a  —  a  cos  0 

are  a  familiar  example  of  the  way  in  which  the  coordinates  of 
points  of  a  curve  may  be  expressed  as  functions  of  a  third  varia- 
ble. In  the  case  of  any  curve,  it  is  obvious  that  this  may  be 
done  in  a  great  variety  of  ways.  Any  tivo  equations  containing 
X,  y,  and  a  that  will  reduce  on  the  elimination  of  a  to  the  ordi- 
nary equation  of  a  given  curve,  can  be  used  as  equations  of  that 
curve. 

For  example  : 


.T  =  2a         1 


y=a+2 


x  =  a  cos  a 


y==asm  a 


x  =  a  cos  a 


are  equivalent  tox  —  2?/-|-4  =  0 


are  equivalent  to  a^  +  ?/^  =  a^ ; 


>  are  equivalent  to  '—  +  -^  =  1  ; 
y  =  biima)  ^*       ^ 


Chap.  X.]  INFINITESIMALS.  153 

ic  =  a  sec  a  1 


=  6tanaJ    "  '"  «'      ^' 


are  equivalent  to  —  —  ^  =  1 , 


The  proof  of  our  proposition  is  as  follows  :  Let  a  and  a  +  zla 
be  the  two  values  of  a  in  question,  and  (ic,?/)  and  {x  +  Jx,  ?/  +  zlz/) 
be  the  two  corresponding  points.  The  distance  D  between  these 
points  will  be,  if  we  use  rectangular  coordinates,  y  {AxY -\- {JyY . 

We  wish  to  prove  that   .   ^q    "T    is  finite. 


D  ^-J{Jxyjt{AyY^   ffWTNM 

and,  by  Art.  149,  ^^  [g]  and  ^^^\  [g]  are  both  finite; 

hence   /™  r^l  ~r    is  finite,  and  D  is  an  infinitesimal  of  the  same 
Ja=0  |_^«J 

order  as  Ja. 

151.  If  two  curves  are  so  connected  that  the  points  of  one  cor- 
respond to  the  points  of  the  other,  so  that  when  a  point  of  the  first 
curve  is  given,  the  corresponding  point  on  the  second  is  determined, 
the  distance  between  two  infinitely  near  points  on  the  first  curve  is 
an  infinitesimal  of  the  same  order  as  the  distance  between  the 
corresponding  x>oints  of  the  second  curve.  For,  if  we  suppose 
the  coordinates  of  the  points  of  the  first  curve  expressed  as 
functions  of  some  variable  a,  the  coordinates  of  the  points  of 
the  second  curve  can  also  be  regarded  as  functions  of  a ;  and, 
b}'  Art.  150,  each  of  the  distances  in  question  will  be  an  infini- 
tesimal of  the  same  order  as  Ja,  and  each  will  therefore  be  ot 
the  same  order  as  the  other. 

152.  If  a  straight  line  moves  in  a  plane  according  to  some  law, 
so  that  each  of  its  positions  corresponds  to  some  value  of  a  varia- 
ble a,  the  angle  between  two  infinitely  near  positions  of  the  line  is 
an  infinitesimal  of  the  same  order  as  the  difference  between  the 
corresponding  values  of  a. 


154  DIFFERENTIAL   CALCULUS.  [Art.   153. 

Suppose  lines  drawn  through  a  fixed  point  O  parallel  to  the 
moving  line  in  its  different  positions.  From  0,  with  the  radius 
unit}',  describe  an  arc.  Consider  any  two  positions  of  the 
moving  line,   and  the  corresponding  lines    at  0,   we  wish  to 


prove  that  the  angle  (f  between  the  latter  is  of  the  same  order 
as  the  difference  between  the  values  of  a  to  which  the  positions 
of  the  moving  line  correspond.  As  all  the  lines  at  0  correspond 
to  values  of  a,  the  points  where  they  cut  the  circle  correspond 
to  values  of  a,  and,  by  Art.  150,  the  distance  AB  between  two 
of  the  points  supposed  to  be  infinitely  near  is  of  the  same  order 

as  Ja.  ^AB  is  equal  to  sin-  ;  therefore  sin^,  and  consequently 
^  itself   is   an  infinitesimal  of  the  same  order  as  Ja,  and  if 

Zf 

limit  [1]  is  finite,    ^^^^^  [-^1  is  finite 


153.  A  simple  geometrical  example  of  an  infinitesimal  of  the 
second  order  is  the  perpendicular  let  fall  upon  the  tangent  at  any 
point  of  a  curve  from  a  second  point  of  the  curve  infinitely  near 
the  first. 

If,  in  our  figure,  the  distance  PP'  is  taken  as  the  principal 
infinitesimal,  PT  is  readil}"  seen  to  be  of  a  higher  order  than 
the  first,  for 

FT       . 

and,  since  c^  =  0  as  P'=  P,  its  sine  =  0  ;  hence 


hmit 
PP'=0 


!T1 

DprJ 


1^1  =  0, 


Chap.  X.]  INFINITESIMALS.  155 


and  P' T  is  an  infinitesimal  of  an  order  higher  than  that  of  I^P\ 
by  Art.  147,  ¥.x. 

To  show  that  P'T  is  of  the  second  order,  let  us  consider  dif- 
ferent secant  lines  drawn  through  P,  PT  being  itself  one  of 
these  lines.  Obviousl}',  each  one  of  these  lines  is  determined 
in  position  when  the  abscissa  of  its  second  point  of  intersection 
with  the  curve  is  given ;  and  therefore  the  angle  between  any 
two  infinitely  near  secant  lines,  as  PP'  and  PT  is  an  infinitesi- 
mal of  the  same  order  as  the  difference  between  the  correspond- 
ing abscissas,  by  Art.  152  ;  but  the  distance  PP'  is  of  the  same 
order,  by  Art.  150  ;  therefore,  <p  and  PP'  are  of  the  same  order, 
that  is,  of  the  first  order  ;  sin^  is  also  of  the  first  order,  by  Art. 
148  ;  hence  P'T,  which  is  equal  to  PP' sin ^,  is  of  the  second 
order  (Art.  147,  Ex.). 

154.  To  determine  the  tangent  at  any  given  point  of  a  curve ^ 
we  draw  a  secant  line  through  the  point  in  question  and  any 
second  point  on  the  curve,  and  seek  the  limiting  position  ap- 
proached by  this  hue  as  the  second  point  approaches  the  first ; 
or,  in  other  words,  we  seek  the  limiting  x>osition  of  the  line  join- 
ing the  given  jyoint  with  an  infinitely  near  point  of  the  curve.  It 
can  be  shown  that  this  is  also  the  limiting  position  of  any  line 
passing  through  the  given  point  and  a  point  whose  distance  from 
the  second  point  of  the  curve  is  an  infinitesimal  of  a  higher  order 
than  the  distance  hetiveen  the  two  points  on  the  curve. 

Let  P  and  P'  be  two  infinitely  near  points  on    p  m 

the  curve,  and  let  P'M  be  an  infinitesimal  of  a 

higher  order  than  PP'.,  then  the  limiting  position 

of  PP'  as  P'=P  will  be  the  same  as  the  limiting  position  of 

PJf ;  for,  in  the  triangle  PJfP', 


156 


DIFFERENTIAL   CALCULUS. 


[Art.  155. 


hence 


ilP'""sin0  ' 


and  as  by  hypothesis,  p^^^     j^ 

ppf_i_A  [sine?]  must  be  zero.     Therefore 

Umit     r   -1     n 
PP'=0  Ln  =  ^' 


0, 


and  the  two  lines,  PP'  and  PJf,  approach  the  same  hmiting 
position. 

155.  This  principle  is  frequently  of  service  in  problems  con- 
cerning the  position  of  tangent  lines.  For  example  :  Supjyose 
perpendiculars  let  fall  from  a  fixed  poiyit  to  the  tangents  of  a  given 
curve,  to  draw  the  tangent  at  any  given  point  of  the  locus  on  ivhich 
the  feet  of  these  perpendiculars  lie. 

Let  M  and  JW  be  two  infinitely  near  points  of  the  given  curve, 
and  0  be  the  given  point  from  which  the  perpendiculars  are  let 
fall ;  then  P  and  P'  are  two  infinitely  near  points  of  the  locus  in 
question,  and  the  required  tangent  at  P  is  the  limiting  position 
of  the  line  joining  P  and  P' .     Draw  through  M  the  line  ilfP" 


parallel  to  the  tangent  M'P'.  If  we  take  MM'  as  our  principal 
infinitesimal,  P"P'  is  an  infinitesimal  of  the  second  order,  by 
Art.  153,  and  PP'  is  of  the  first  order,  by  Art.  151  ;   conse- 


Chap.  X.]  INFINITESIMALS.  157 

quentl}'  (Art.  154)  it  will  answer  our  purpose  to  find  the  limit- 
ing position  of  the  line  joining  PP"  ;  but,  since  MP"0  and  MPO 
are  both  right  angles,  P"  lies  on  the  circumference  of  a  circle 
described  on  OM  as  diameter,  and  the  required  limiting  position 
of  PP"  is  that  of  a  tangent  to  this  circle  at  P,  which  is  therefore 
the  required  tangent.  Hence  to  obtain  a  tangent  to  the  locus  in 
question  at  an}'  given  point,  we  have  only  to  join  the  correspond- 
ing point  with  0,  to  erect  a  circle  on  this  joining  line  as  diameter, 
and  to  draw  a  tangent  to  the  circle  at  the  given  point.  Of  course, 
the  normal  to  this  locus  at  the  given  point  bisects  the  joining 
Kne  OM. 

156.  Let  us  consider  the  locus  of  the  feet  of  peipendiculars  let 
fall  from  the  focus  of  an  ellipse  upon  the  tangents  to  the  curve. 

Since  the  tangent  to  the  required  locus  at  P  is  tangent  to  the 
circle  on  FM  as  diameter,  the  normal  at  P  passes  through  the 


centre  C  of  the  circle.  Draw  the  focal  radius  F^M.  Since  the 
tangent  to  an  ellipse  makes  equal  angles  with  the  focal  radii 
drawn  to.  the  point  of  contact, 

TMF^=PMC\ 

PMC  =  MPC, 

because  MC  and  CP  are  equal ; 

.'.  MPC=TMFi, 

and  PO  is  parallel  to  MF^ ;  it  must  then  divide  MF  and  F^F 
proportionally ;  and  as  it  bisects  MF,  it  also  bisects  F^  F,  and 


158 


DIFFERENTIAL   CALCULUS. 


[Art.  157. 


consequently  passes  through  the  centre  of  the  ellipse.  Since  every 
normal  to  the  required  locus  passes  through  the  centre  of  the 
ellipse,  the  locus  is  a  circle  concentric  with  the  eUipse.  It  is 
easily  seen  that  it  must  pass  through  the  vertices  of  the  ellipse. 
It  is  then  a  circle  on  the  major  axis  of  the  ellipse  as  diameter. 

Example. 

Show  that  the  locus  of  the  foot  of  a  perpendicular  let  fall  from 
the  focus  upon  any  tangent  is  a  circle  on  the  transverse  axis  as 
diameter  in  the  hyperbola ;  is  the  tangent  at  the  vertex  in  the 
parabola. 

Problem. 

157.  Upon  each  normal  to  a  plane  curve  a  point  is  taken  at  a 
constant  distance  from  the  intersection  of  the  normal  ivith  the 
curve;  to  find  the  tangent  at  any  point  of  the  locus  thus  formed. 

Let  M  and  M'  be  two  infinitely  near  points  on  the  given  curve, 
P  and  P'  the  corresponding  points  of  the  locus  ;  let 

MP=M'F'=a; 

call  the  angle  between  the  normals,  <p.  Draw  MM"  and  PP" 
perpendicular  to  the  second  normal.     The  required  tangent  is  the 


limiting  position  of  PP'^  and  the  tangent  at  M  is  the  limiting 
position  of  MM'.  If  MM'  is  taken  as  the  principal  infinitesimal, 
PP'  and  <p  are  of  the  first  order  and  M'M"  of  the  second  (Arts. 
151-153) .     P'F"  is  of  an  order  higher  than  the  first,  for 


Chap.  X.]  INFINITESIMALS.  159 

P'P"=  M'M"+  M"P"-  a, 
M"P"=acos<p', 
hence  P'P"=  M'3f" -  a{l- coscp). 

0  being  of  the  first  order,  1  —  cos  <f  is  of  the  second  order  by 
Art.  148  ;  and  as  31' M"  is  of  the  second  order,  P'P"  is  of  at 
least  as  high  an  order  as  the  second.  Bj-  Art.  154,  our  required 
tangent  will  be  the  limiting  position  of  PP",  and  the  tangent  at 
M  will  be  the  Hmiting  position  of  M3I"  ;  but  PP"  and  MM"  are 
parallel  alwaj^s ;  therefore  their  limiting  positions  are  parallel, 
and  our  required  tangent  is  parallel  to  the  tangent  to  the  given 
curve  at  the  corresponding  point,  and  the  curves  are  what  are 
called  parallel  curves. 

Problem. 

158.  An  angle  of  constant  magnitude  is  circumscribed  about  a 
given  curve;  to  draio  a  tangent  to  the  locus  of  its  vertex. 

The  required  tangent  is  the  limiting  position  of  the  secant 
line  PP'.  Draw  through  31  and  N  hues  MP",  iVP",  parallel 
to  the  tangents  at  M'  and  N'.  It  can  be  shown  that  the  sides, 
and  therefore  the  diagonal,  of  the  parallelogram  P'P'^  are  in- 
finitesimals of  a  higher  order  than  PP',  and  therefore  that  the  re- 

P' 


quired  tangent  can  be  found  as  the  limiting  position  of  PP".  Since 
the  angles  at  P  and  P"  are  equal,  the  point  P"  lies  on  a  circle  cir- 
cumscribed about  MPN\  the  limiting  position  of  PP"  is  there- 


160  DIFFERENTIAL  CALCULUS.  [Art.  159. 

fore  the  tangent  to  this  circle  at  P.  Our  solution  is,  then,  draw 
a  circle  through  the  vertex  of  the  circumscribing  angle  and  the 
points  of  contact  of  its  sides,  and  the  tangent  to  this  circle  at 
the  vertex  of  the  angle  is  the  tangent  required. 

Example. 

Show  that  the  locus  of  the  vertex  of  a  right  angle  circum- 
scribed about  an  ellipse  or  an  h}  perbola  is  a  concentric  circle ; 
about  a  parabola  is  the  directrix. 

159.  In  the  preceding  examples,  the  advantage  we  have 
gained  in  the  use  of  infinitesimals  has  arisen  from  the  fact  that 
we  have  been  able  to  replace  one  infinitesimal  b}-  another  related 
to  it  and  more  simpl}-  connected  with  the  other  values  consid- 
ered in  the  problem.  The  possibility  of  such  substitutions,  and 
the  limitations  under  which  they  can  be  made,  form  the  subject 
of  the  following  two  theorems,  which  are  of  prime  importance, 
and  lie  at  the  foundation  of  the  Infinitesimal  Calculus. 


Theorem. 

160.  In  any  problem  concerning  the  limit  of  the  ratio  of  two 
infinitesimals^  either  may  be  replaced  by  any  infinitesimal  so  related 
to  it  that  the  limit  of  the  ratio  of  the  second  to  the  first  is  unity. 

Proof. 
Let  a,  /?,  a',  and  /?'  be  infinitesimals  so  related  that 


. .  a 


:„u/S' 


limit  -  =  1  and  limit  -  =  1. 


Then  will  limit-  =  limit" 


a       a'     a     /?' 


^     ^    ^  identically ; 


y?       /?'    a'    y? 


Chap.  X.] 
hence 


INFINITESIMALS. 


limit- =  limit—  x  limit ^  X  limit—, 

l3  /5'  «'  /S 


limit  -  =  limit—  x  1  X  1  =  limit—. 


/5' 


161 


Q.E.D. 


Theorem. 

161.  In  any  problem  concerning  the  limit  of  a  sum  of  infini- 
tesimals^ jv'ovidecl  that  this  limit  is  finite,  any  infinitesimal  may 
be  replaced  by  another  so  related  to  it  that  the  limit  of  the  ratio  of 
the  second  to  the  first  is  unity. 


Let 


Proof. 


be  a  sum  of  infinitesimals  of  such  a  nature  that  the  number  of 
the  terms  increases  as  each  term  decreases  in  absolute  value,  so 
that  the  limit  of  the  sum  is  some  finite  quantitj'. 

Let  /?i,  ^21 1%^ i^n  ^6  ^  set  of  infinitesimals  so  related  to  the  first 

set  that         limit— =  1,  limit— =  1,  &c.,  limit— =1, 


then 


"1  "2  ^n 

£i,£2i c„  being  necessarily  infinitesimal  (Art.  7). 

/^2=  '''2  4-«2^2» 

t^l  -f  i\  +  /^3  + +  l\  =  «1  +  «2  +  «3  + -f-  «H 

-|-«iei4-  «2^2  4    "3^3+ +^^n'n- 


162  DIFFERENTIAL   CALCULUS.  [Art.  162. 

Let  1]  be  such  a  ^'ariable  that  at  any  instant  it  shall  be  equal  to 

the  greatest  in  absolute  value  of  the  quantities  ei,£2, £„.     Of 

course,  since  each  of  these  approaches  zero  as  its  limit,  >y  must 
also  approach  zero  as  its  limit ;  i.e.,  t]  is  infinitesimal. 

«l^l  +  «2-2-f-«3^3-f- +«„^„<>?(«l+«2H-«3+ +^n)5 

hence    h-^h^-h^- +  ^„ -  («i  +  «2  +  «3  + +  «„) 

<'/(«l  +  «2+«3+ +  «»)• 

By  hypothesis,  limit  {a'^-\- ao-\- +  ««)  is  finite  ; 

therefore,  limit  of  ly  (a,  +  «2  +  «3  + +  ««)  is  zero. 

Consequently 
nmit(/3i  +  /'5,  +  /^3+ 4  /\)  =  limit  (ai  +  «2-f-a3  + -haj. 

Q.E.D. 

162.    If  tioo  infinitesimals  differ  from  each  other  by  an  infini 
tesimal  of  a  higher  order ^  the  limit  of  their  ratio  is  unity. 


For, 

let 

a' —  a  =  £^ 

where 

£  is 

of  a 

higher  order  than  a  ; 

a                a 

limit- =1  + limit-; 

a  a 

but,  by  hypothesis,  limit- =  0,  (Art.  147,  Ex.); 


a 


therefore  limit —  =  1 , 

a 


Chap.  X.] 


INFINITESIMALS. 


163 


It  follows  that  the  theorems  of  Art.  160  and  Art.  161  can  be 
stated  as  follows  :  — 

In  finding  the  limit  of  a  ratio ^  or  the  limit  of  a  sum  of  infini- 
tesimals^ any  infinitesimal  may  he  replaced  by  one  that  differs 
from  it  by  an  infinitesimal  of  a  higher  order.  Or,  in  finding 
the  limit  of  a  ratio  or  of  a  sum  of  infinitesimals^  any  infinitesimal 
term  may  be  neglected  without  in  the  least  affecting  the  restdt,  pro- 
vided that  it  is  of  a  higher  order  than  the  terms  retained. 

163.  Let  us  take  the  problem  of  finding  the  direction  of  the 
tangent  to  a  parabola. 

The  tangent  T'T  at  P  is  the  limiting  position  of  the  secant 
through  P  and  P'.  Draw  the  focal  radii  FP  and  FP',  and  the 
perpendiculars  PE  and  P'S  to  the  directrix.     Draw  PM  and 


PN  perpendicular  to  FP'  and  P'S  respectively,  and  with  F  as  a 
centre,  and  with  the  radius  FP^  describe  the  arc  PQ. 

Take  PP'  as  the  principal  infinitesimal,  then  P'M  and  P'K  are 
of  the  first  order,  since  the  limit  of  the  ratio  of  each  of  them  to 
PP^  is  finite. 

PQ  is  of  the  first  order,  by  Art.  151,  and  MQ  is  of  the  second 

order,  by  Art.  153. 

P'S=P'F, 

from  the  definition  of  a  parabola ; 

PIi=PF=:QF', 

.-.  P'N=P'Q. 
P'M 


cosPP'F: 


PP'' 


164  DIFFERENTIAL   CALCULUS.  [Art.  164. 

P'N 


cosPP'S  =  -^, 


cos  T'FF^  limit  [ cos  PP'F' 
cos  T'PR     P'=p[_cosPP'^ 


limit 
P 


_  limit  r^ 

-P'=PIP' 


imit   [PlMl 
'  =  p\_P'J^j 

1,  by  Art.  162; 


.-.   T'PR=T'PF, 

and  the  tangent  at  an}-  point  of  a  parabola  bisects  the  angle 
between  the  focal  radius  and  the  diameter  through  the  given 
point. 

164.  To  find  the  area  of  the  sector  of  a  parabola  included  be- 
tween two  focal  radii.  Take  points  of  the  parabola  between  the 
extremities  of  the  bounding  radii,  and  join  them  with  the  focus, 
thus  dividing  the  area  in  question  into  smaller  sectors,  of  which 
the  sector  FPP'  in  the  figure  of  the  last  article  may  be  taken  as 
a  type.  Draw  perpendiculars  from  the  extremities  of  the  bound- 
ing radii  to  the  directrix,  and  consider  the  external  area  bounded 
by  them,  the  directrix  and  the  curve  ;  draw  perpendiculars  from 
the  intermediate  points  already  described  to  the  directrix,  and  the 
external  area  will  be  divided  into  smaller  curvilinear  quadrilate- 
rals, of  which  PP'ES  is  one.  No  matter  how  close  together  the 
intermediate  points  are  taken,  the  external  area  is  the  actual  sum 
of  these  small  curvilinear  quadrilaterals ;  it  is  then  the  limit  of 
their  sum  as  the  number  is  indefinitely  increased.  If  the  distance 
between  any  two  of  the  points,  as  PP',  is  taken  as  the  principal 
infinitesimal,  FN,  P'N,  PM,  P'M,  are  all  infinitesimals  of  the 
first  order,  since  the  limit  of  the  ratio  of  each  of  them  to  PP' 
is  the  sine  or  the  cosine  of  a  finite  angle.  The  area  of  PP'MS 
lies  between  P'S  X  PJSf  and  NS  X  PJSf,  and  is  therefore  an  infini- 
tesimal of  the  first  order.  Hence  we  have  to  consider  the  limit 
of  a  sum  of  infinitesimals  where  the  limit  is  finite,  and  we  can 
replace  an}^  one  by  one  differing  from  it  by  an  infinitesimal  of  a 
higher  order  than  the  first.     The  rectangle  FESJSf  differs  from 


Chap.  X.j  INFINITESIMALS.     >  165 

PP'RS  by  less  than  a  rectangle  on  FN  and  P'jV;  that  is,  by  less 
than  PNx  P'N^  an  infinitesimal  of  the  second  order.  Therefore 
the  required  external  area,  which  is  the  limit  of  the  sum  of  infini- 
tesimal areas  of  which  PP'RS  is  a  type,  and  which  we  shall  indi- 
cate b}'  limit  2.PP'RS  {I,  serving  as  a  symbol  for  the  word  sum), 
is  equal  to  limit  2.PENS. 

The  given  sector  is  equal  to  the  sum  of  the  smaller  sectors  of 
which  FPP'  is  a  t3'pe  =  limit  I^FPP' ,  each  term  here  being  an 
infinitesimal  of  the  first  order.  Draw  the  straight  line  PQ.  The 
triangle  FPQ  diflfers  from  the  sector  FPP'  by  less  than  a  rect- 
angle on  PM  and  MP\  which  would  be  of  the  second  order,  and 
ma}'  therefore  replace  FPP^  in  the  expression  for  our  required 
area. 

limit  ^=1,  by  Art.  163; 

consequently  PM  and  PN  diflfer  by  an  infinitesimal  of  higher 
order  than  the  first,  and  the  triangle  FPQ  diflTers  from  one-half 
the  rectangle  PENS  hy  an  infinitesimal  of  higher  order  than  the 
first,  and  may  be  replaced  hy  ^PRNS. 

We  have  then,  external  area  =  \\m\t ZPRNS^ 

given  sector  =  \\m\il^ PRNS  ; 

and  the  given  focal  sector  is  equal  to  one-half  the  area  bounded 
bj"  the  curve,  the  directrix  and  perpendiculars  let  fall  from  the 
extremities  of  the  arc  of  the  given  sector  to  the  directrix. 

Infinitesimal  Arc  and  Chord. 

165.  Let  us  consider  the  relation  between  the  lengths  of  an 
infinitesimal  chord  and  its  arc. 

Take  the  chord  PP'  as  the  principal  infinitesimal,  and  draw 
the  tangents  PT  and  P'T.     The  arc  PP'  is  ^ 

less  than  PT-\-P'T  and  greater  than  the 
chord  PP'.  The  angles  e  and  s'  are  infini- 
tesimal. 


166  DIFFERENTIAL  CALCULUS.  [Art.  166. 


PM 

cos;  =  — — , 
PT 

-.■=£f. 

limit  cos£  =  l, 

and 

limit  cos£'=l  ; 

therefore 

PT 

and 

limit^;^=l. 
P'T 

Thus 

PM=PT+ri 

and 

P'M=P'T+r/, 

where  rj  and  r/  are  infinitesimals  of  a  higher  order  than  the  first, 
by  Art.  147,  Ex. 

PM+  P'M=  PT+  P'T-{-  Tj  +  ri\ 

or  the  difference  between  the  sum  of  the  tangents  and  the  chord 
is  of  a  higher  order  than  the  first.  The  difference  between  the 
arc  and  the  chord  is  less  than  this,  therefore  the  limit  of  the  ratio 
of  an  infinitesimal  arc  to  its  chord  is  unity. 

166.  It  is  customary  to  say  ro\tghly  that  lines  which  make  with 
each  other  an  infinitesimal  angle,  that  is,  lines  which  approach 
the  same  limiting  2)osition,  coincide,  and  that  finite  values  which 
differ  by  an  infinitesimal  or  infinitesimal  values  which  differ  by 
an  infinitesimal  of  a  higher  order,  that  is,  values  such  that  the 
limit  of  their  ratio  is  unity,  are  equal;  and  this  wa}^  of  speaking 
is  very  convenient,  especially  for  preliminar}-  investigations.  It 
is  important,  however,  to  be  able  to  put  a  proof  given  in  this 
form  into  the  more  exact  language  of  limits. 

It  is  easily  seen  from  what  has  just  been  said,  that  the  line 


Chap.  X.]  INFINITESIMALS.  167 

joining  tico  infinitely  near  points  of  any  curve,  can,  speaking 
roughly,  be  regarded  at  pleasure  as  chord,  arc,  or  tangent,  so  that 
an  infinitesimal  arc  can  he  treated  as  a  straight  line. 

167.  As  an  example  of  this  loose  form  of  proof,  let  us  show 
that  a  tangent  to  an  ellipse  makes  equal  angles  with  the  focal 
radii  drawn  to  the  point  of  contact. 

Let  P  and  F'  be  two  infinitely  near  points  of  the  ellipse,  then 
PP'  is  the  tangent  in  question.    From  F  and  F'  as  centres,  draw 


the  arcs  PA  and  P'B  ;  PA  and  P'B  being  infinitesimal  arcs,  are 
straight  lines,  and  PAP'  and  P'BP  are  right  angles,  since  the 
tangent  to  a  circle  is  perpendicular  to  the  radius  drawn  to  the 
point  of  contact. 

F>P+PF=F'P'-\-P'F, 

by  the  definition  of  an  ellipse.  Take  away  from  the  first  sum 
F'P  +  BF,  and  we  have  left  PB ;  take  away  from  the  second 
sum  the  equal  amount  F'A  +  P'F,  and  we  have  left  PA  ; 

.•.PB=P'A; 

and  the  right  triangles  PAP'  and  PBP'  have  the  h3'pothenuse 
and  a  side  of  the  one  equal  to  the  hj^iothenuse  and  a  side  of  the 
other,  and  are  equal ;  and  the  angle 

FPP'=  F'P'P ; 

but  the  lines  F'P'  and  F'P  coincide,  so  that  the  angle  F'P'P  is 
the  same  as  the  angle  F'PT ;  and 

.-.  F'PT=FPP', 

and  the  tangent  makes  equal  angles  with  the  focal  radii,    q.e.d. 


168 


DIFFERENTIAL   CALCULUS. 


[Art.  168. 


P^XAMPLE. 

Prove  that  a  tangent  to  an  hj'perbola  bisects  the  angle  between 
the  focal  radii  drawn  to  the  point  of  contact. 

168.  To  find  the  area  of  a  segment  of  a  parabola  cut  off  by  a 
line  perpendicular  to  the  axis.  Compare  the  required  area  with 
the  area  of  the  circumscribing  rectangle.     We  can  regard  the 


Q 

N 

V 

^^^^-'^^ 

A 

^ 

R 

^ 

\r 

;// 

P 

y 

/ 

^ 

TT) 

"0 

F 

s 

u 

first  as  made  up  of  the  infinitesimal  rectangles  of  which  PMUS 
is  a  t;y'pe,  and  the  second  of  the  corresponding  rectangles  of 
which  QNPK  is  one.     Draw  the  directrix. 

FF=  SD  ami  DO  =  OF, 

by  the  definition  of  the  parabola  ;  but 

PF=FT  by  Art.  163; 

.-.  T0=  OS. 

The  triangles  P'MP  and  PST  are  similar,  and 

P'M^  PM^  PM . 
PS       ST      20S' 

hence  PM  xPS=20Sx  P'M=  2PEQN, 

or  rectangle  PU=  2PQ  ; 

.'.2PLr=22:PQ, 


Chap.  X.] 


INFINITESIMALS. 


169 


and  the  segment  in  question  is  twice  the  external  portion  of  the 
circumscribing  rectangle,  and,  therefore,  is  two-thirds  of  the 
whole  rectangle. 

Example. 

Prove  the  theorems  of  Arts.  167,  168,  strictl}^,  bj'  the  method 
of  limits. 


169.  The  properties  of  the  C3'cloid  can  be  very  simply  and 
neatly  obtained  b}^  the  aid  of  infinitesimals  ;  though,  for  this  pur- 
pose, it  is  better  to  look  at  the  curve  from  a  new  point  of  view. 

Let  a  fixed  circle  equal  to  the  generating  circle  be  drawn  tan- 
gent to  the  base  of  the  cycloid  at  its  middle  point ;  through  the 


generating  point  P,  draw  PQD  parallel  to  the  base.  From  the 
nature  of  the  C3'cloid,  the  arc 

PN=  ON  and  OB  =  ACB, 

Pq  =  NB=OB-  0N=  ACB -QB  =  ACQ. 

Hence  j^ohits  of  the  cycloid  can  be  obtained  by  erecting  perpendicu- 
lars to  a  diameter  of  a  fixed  circle^  and  extending  each  until  its 
external  portion  is  equal  to  the  distance  along  the  arc  of  the  circle 
from  the  perpendicular  in  question  to  a  given  end  of  the  diameter. 


170.    The  tangent  to  the  cycloid  passes  through  the  highest  point 
of  the  generating  circle. 


170 


DIFFERENTIAL   CALCULUS. 
T 


[Art.  170. 


Rough  Proof.  —  Let  P  and  P'  be  infinitely  near,  then  PP'  is 
the  required  tangent ;  through  P'  draw  an  arc  parallel  and  simi- 
lar to  QQ' .  This  arc  maj'  be  regarded  as  a  straight  line.  The 
triangle  PPM  is  isosceles,  since 

qp=  ACQ  and  MQ  =  P'Q'=  ACQ', 

hence  P3I=  QQ'=  MP' ; 

.  • .  the  angle  PP'M  =  P'PM. 

P'M  is  parallel  to  the  tangent  at  P  to  the  generating  circle,  hence 

PP'M=  TPT\ 

and  PT  bisects  the  angle  MPT' ^  bisects  the  arc  PTS^  and  con- 
sequently passes  through  the  highest  point  of  the  generating 
circle.  q.e.d. 

Strict  Proof.  —  Draw  the  chord  P'M^  and  regard  PP'  as  a 
secant  line  ;  in  the  triangle  PP'M  w^e  have 

sXnPP'M     PM 


The 


sin  TPM      P'M' 

,.    .^  sin  PP'Jf     T    .,  PM 

limit =  limit  — ; — 

sin  TPM  P'M 

arc  P'M=PM, 


and  the  chord  P'M  differs  from  the  arc  b}'  an  infinitesimal  of  a 
higher  order  than  that  of  the  chord. 


Chap.  X.] 


INFINITESIMALS. 


171 


.*•.  limit 


PM 


=  limit =rT^-_=  1 , 


chord  P'M  arc  P'M 

hence  limit  PP'M==  limit  TPM. 

The  limiting  position  of  P'M  is  the  tangent  PT' ; 

,-.  limit  TPQ  =  limit  TPT', 

and  the  tangent  passes  through  the  highest  point  of  the  generat- 
ing circle. 

The  Area  of  the  Cycloid. 

171.  Bough  Investigation. — Circumscribe  a  rectangle  about 
the  cycloid,  and  its  area  is  evidently  equal  to  the  circumference 
of  the  generating  circle  multiplied  by  its  diameter ;  that  is,  to 
four  times  the  area  of  the  circle.     The  area  of  the  cjcloid  is 


N  a 

b         T 

A 

/ 

z 

?-<r 

\^ 

/ 

\ 

/ 

c 

A 

/             J 

' 

K 

/' 

V 

0 

^^ 

^^ 

this  area  minus  the  area  of  the  external  portion  of  the  rectangle. 
The  external  area  ANO  ma}^  be  divided  into  trapezoids,  of  which 
ahPP'  is  any  one.  The  tangent  PP'  passes  through  the  highest 
point  of  the  generating  circle,  and  is  a  diagonal  of  the  rectangle 
TajPc,  Tc  being  a  diameter.     From  geometry, 

abP'P=cdP'P, 

which  is  equal  to  Qg  ;  therefore  the  sum  of  the  trapezoids  abP'P 
is  equal  to  the  sum  of  the  corresponding  rectangles  Qg^  or  the 
external  area  AJSfO  is  equal  to  the  semi-circle  ACB :  but  ANO 


172 


DIFFERENTIAL   CALCULITS. 


[Art.  172. 


is  half  of  the  external  portion  of  the  circumscribing  rectangle ; 
consequently,  the  area  of  the  cycloid  is  three  times  the  area  of 
the  generating  circle. 

Strict  Proof.  —  The  external  area  is  the  sum  of  the  curvilinear 
quadilaterals  of  which  ahP'P  is  any  one  ;  that  is, 

area  =  labP'P=  limitlabP'P=  limitlabhP, 

for  abhP -  abP'P< eP'hP, 

which  is  of  the  second  order.    P'P"  is  of  the  second  order,  since 

N  a       h'b     T 


it  is  proportional  to  the  distance  from  P'  to  the  tangent  at  P 
(Art.  153);  therefore  bhhV  is  of  the  second  order,  and 

limit  Ja6/iP=  limit  Ja&7i'P. 

ab'h'P=edcP=Qg, 

hence         the  external  area  =  limit  IQg  —  area  of  ACB. 


Length  of  an  Arc  of  the  Cycloid. 

172.  Rough  Proof.  — The  arc  AP  is  equal  to  the  sum  of  the 
infinitesimal  chords  of  which  PP'  is  one.  The  chord  AQ'i^  the 
sum  of  the  differences  between  each  chord  and  the  one  drawn  to 
a  point  of  the  fixed  circle  above  the  point  in  question  and  in- 
finitely near  it ;   QS  is  such  a  difference,  hence 

ai'C^P  =  2PP'  and  chord^lQ  =  ^QS. 


Chap.  X.] 


INFINITESIMALS. 


178 


PP'  and  QR  are  equal,  Q'QR=  Q'RQ, 


by  Art.  170, 


and  QQ'R  is  isosceles.  Q'S^  an  infinitesimal  arc  described  from 
^  as  a  centre,  ma}'  be  regarded  as  a  straight  line  perpendicular 
to  QR^  and  therefore  bisects  QR^  and 

PP'=2QS, 

IPP'=  21 QS. 

ArcAP=2'chordAQ, 

The  arc  ^10  =2^5, 

and  the  ivliole  arc  of  the  cycloid  is  eight  times  the  radius  of  the 
generating  circle. 

Strict  Proof.  —P'P",  Q'T",  and  US  are  infinitesimals  of  the 
second  order,  each  being  proportional  to  the  distance  from  a  point 


of  a  curve  to  the  tangent  at  a  point  infinite!}'  near.     Vv  is  also 
of  second  order,  as  it  is  the  projection  of  Q'T"  on  AQ. 


174  DIFFERENTIAL   CALCULUS. 

The  arc  AP=  limit2'PP'=  limitl'PP", 

since,  in  triangle  PP'P'\ 

PP'-PP"<P'P". 
The  chord  AQ  =  IQS', 

=  limit  IQS  =  limit  IQU =\imit  IQ  V. 
But  the  triangle  QT"R  is  isosceles,  hence 
QV=hQR  =  ^PP"', 
and,  as  arc  AP  ==  limit  IPP" , 

arc  ^P=  2  chordae. 


[Art.  173. 


Q.E.D. 


Radius  of  Curvature  of  the  Cycloid. 

173.  Rough  Investigation.  —  The  centre  of  curvature  for  P  is 
the  intersection  of  the  normal  at  P  with  the  normal  at  P' . 

PX,  P'X,  and  PP'  are  parallel  to  QB,  Q'B,  and  QS  respec- 
tively, hence  the  triangles  PP'X  and  QSB  are  similar.    The  angle 


Q  is  a  right  angle,  the  angle  B  is  infinitesimal ;  the  angle  QSB 
differs  from  a  right  angle  by  an  infinitesimal,  and  may  be  re- 
garded as  a  right  angle.     Therefore,  by  Art.  172, 

QS=^PP\ 


Chap.  X.] 

and  consequently, 


INFINITESIMALS. 


175 


and  the  radius  of  curvature  is  twice  PN,  the  portion  of  the  normal 
within  the  generating  circle. 

Strict  Proof.  —  The  centre  of  curvature  is  the  limiting  posi- 
tion of  X. 

T  A 


PP"X'  is  similar  to  QRB,  hence 


PX'  PP"  ,   ,.       .,    PX'  y       ..    PP' 

and  limit  —-—  =  limit 


Q^     Q,n 


QB 


QR 


(1) 


Let  PP'  be  the  principal  infinitesimal,  then  P'P"  is  of  the  second 
order  ;  therefore,  in  (1),  PX  can  be  substituted  for  PX\    RQ' S 


IS 


similar  to  BQR,  hence  =  — — . 

QR     QB 


QR  and  QS  are  infinitesimal,  QB  is  finite,  RS  is  of  the  second 
order,  and  QS  can  be  substituted  for  QR  in  (1),  and 


PY  PP" 

limit  ^  =  limit 


but,  by  Art.  172, 


QB 

PP" 
hmit  ^-^  =  2 
QS 


QS 


• .  limit  PX  =2QB  =  2PN. 


Q.E.D. 


176 


DIFFERENTIAL   CALCULUS. 


[Art.  174. 


Evolute  of  the  Cycloid. 
174.    Extend  the  diameter  TN  to  N\  making 

NN'=  TN 

and  draw  a  circle  on  NN'  as  diameter.     The  centre  of  curva- 
ture X,  corresponding  to  P,  will  lie  on  this  circle,  since 


Draw  a  tangent  to  the  second  circle  at  N',  drop  a  perpendicular 
from  0  to  this  tangent,  and  lay  off  B'O'  equal  to  one-half  the 
circumference  of  the  generating  circle. 


The 


sircP]Sr=ON=B'N'; 
.-.  the8ircN'X=N'0', 


and  X  lies  on  a  C3'cloid  equal  to  the  given  cycloid,  having  its 
origin  at  0'  and  its  highest  point  at  0,  and  this  must  be  the 
evolute  required. 


Chap.  X.]  INFINITESIMALS.  177 

Examples. 

175.  (1)  From  a  point  0  situated  in  the  plane  of  a  plane 
curve,  radii  vectores  are  drawn  to  different  points  of  the  curve, 
and  on  each  one  a  distance  is  laid  off  from  0  inversely  propor- 
tional to  the  length  of  the  radius  vector ;  to  determine  the  tan- 
gent at  any  point  of  the  locus  of  the  points  thus  obtained. 

(2)  Take  an}'  two  curves  in  the  same  plane,  and  consider  as 
corresponding  points  those  at  which  the  tangents  are  parallel ; 
draw  through  a  fixed  point  lines  equal  and  parallel  to  those 
uniting  corresponding  points  of  the  two  curves.  Prove  that  a 
tangent  to  the  locus  of  the  points  thus  obtained  is  parallel  to  the 
tangents  at  the  corresponding  points  of  the  given  curves,  and 
that  any  arc  of  this  curve  is  the  sum  or  difference  of  those  which 
correspond  to  it  upon  the  given  curves. 

(3)  From  a  point  0  radii  vectores  are  drawn  to  a  given  curve, 
and  each  is  extended  be3'ond  the  curve  by  a  constant  length. 
Prove  that  the  normal  to  the  curve  on  which  the  extremities  of 
the  radii  vectores  lie,  the  normal  at  the  corresponding  point  of 
the  given  curve,  and  the  perpendicular  through  0  to  the  radius 
vector  of  the  point,  have  common  intersection. 

176.  To  show  the  power  of  this  method  of  infinitesimals,  we 
shall  give  an  investigation  into  the  nature  of  what  is  called  the 
Braclustochrone^  or  Curve  of  Quickest  Descent.  The  problem  is 
a  famous  one,  and  the  solution  below  is  in  effect  the  one  given 
by  James  Bernouilli,  and  is  very  much  simpler  and  more  ele- 
mentary than  the  usual  analytical  solution  which  requires  the 
use  of  the  Calculus  of  Variations. 

The  problem  is,  given  two  points  not  in  the  same  horizontal 
plane.,  nor  in  the  same  vertical  line;  to  find  the  curve  doivn 
which  a  particle  moving  tvithout  friction  can  slide  in  the  least  time 
from  the  upper  point  to  the  lower,  the  accelerating  force  being 
terrestrial  gravitation. 

Let  us  first  consider  a  simpler  question :  To  find  the  path  of 
quickest  descent  on  the  hypothesis  that  it  is  to  consist  of  two 


178 


DIFFERENTIAL   CALCULUS. 


[Art.  176. 


straight  lines  intersecting  on  a  givenhorizontalplane^ 
assuming  that  the  particle  moves  down  each  line 
with  a  uniform  velocity  equal  to  the  mean  velocity 
with  which  it  would  actually  descend  the  line  in 
question.  It  is  easil}"  seen  that  both  lines  must 
lie  in  the  vertical  plane  containing  the  two  given 
points. 

Let  PNP'  and  PMP'  be  two  paths  of  equal  time 
from  P  to  P'.  Then  the  required  path  must  lie 
between  them.  If  we  suppose  them  to  approach, 
continuing  still  paths  of  equal  time,  the  required 
path  of  quickest  descent  will  be  the  limiting  posi- 
tion of  either  of  them.  Let  v  be  the  mean  velocity 
of  a  particle  sliding  from  Pio  M\  then,  b}'  Art. 
115,  -y  will  also  be  the  mean  velocity  of  a  particle 
sliding  from  P  to  N. 
Let  Vi  be  the  mean  velocity  of  a  particle  sliding  from  M  to  P', 
supposing  that  the  particle  started  from  M  with  the  velocity 
actuall}'  acquired  b}'  shding  down  PM\  then  Vi  is  also  the  mean 
velocity  of  descent  from  Nio  P',  by  Art.  115.  As  we  are  going 
to  make  the  paths  PMP'  and  PNP^  approach  indefinitely,  MN 
is  an  infinitesimal.  Draw  the  arcs  NS^  and  MR'  from  P  and  P' 
as  centres,  and  the  perpendiculars  NS  and  MR,  On  our  hy- 
pothesis, the  time  of  descent  from  P  to  S'  equals  time  of  descent 
from  P  to  A^,  and  time  of  descent  from  M  to  P'  equals  time  of 
descent  from  P'  to  P';  hence,  as  time  PMP'  equals  time  PNP\ 
the  time  of  descent  from  S'  to  M  equals  time  fronii  A^  to  R\ 


or 


whence 


and 


S'M 

NR' 

V 

'Vi 

S'M 
NR' 

_  V 

""'•II- 

=  limit  —  ; 

■'-11^ 

limit 

SM 
JS-R' 

Chap.  X.] 


INFINITESIMALS. 


1T9 


hence 


JSM_  cos PMN  _  cosy  . 
NR  ~~  cos  P'MN  ~  cos  y i ' 


cos  9?        ,      .    V 
limit =  limit  — 

cos  (fi  Vi 


Let  the  angles  made  with  the  horizontal  by  the  two  portions  of 
the  required  path  be  d  and  (9i,  and  the  mean  velocities  down  the 
two  portions  of  the  required  path  be  v  and  Vi.     Then 


cos  0      V         cos  6  _  cos  01 


77—  ~  or 

COS  ^1        Vi 


V-i 


Let  us  now  consider  a  path  of  quickest  descent,  consisting  of 
three  rectilinear  powtions  intersecting  on  given 
horizontal  planes,  all  the  other  conditions 
remaining  as  before.  Let  PMSP'  be  the 
required  path.  It  is  easil}'  seen  that  PBS 
must  be  the  path  of  quickest  descent  under 
the  given  conditions  from  P  to  S ;  so  that 

cos  d  _  cos  01 

V      "      Vi     ' 

ESP'  must  be  the  path  of  quickest  descent  from  R  to  P'  under 
the  given  conditions,  so  that 

cos  ^1      cos  0q 


Vo 


V,  -Ui,  V2  being  mean  velocities  down  PR,  RS,  and  SP\  respec- 
tively. 

Suppose  now  that  the  number  of  rectilinear  portions  of  the 
broken  line  of  descent  is  indefinitel}^  increased,  each  portion  will 
decrease  indefinitely  in  length,  and  the  path  will  approach  a 
curve  as  its  limiting  form.  The  mean  velocity  down  each  por- 
tion of  the  polygonal  path  will  approach  as  its  limit  the  actual 
velocity  at  the  corresponding  point  of  the  limiting  curve ;  the 
angle  made  by  each  portion  with  the  horizontal  will  approach  the 
angle  made  bj-  the  curve  with  the  horizontal :  hence  our  limit- 


180 


DIFFERENTIAL   CALCULUS. 


[Art.  176. 


ing  curve,  which  is  obviously  the  required  brachistochrone^  must 
be  of  such  a  nature  that  the  cosine  of  the  angle  it  snakes  at  each 
point  with  the  horizontal  shall  he  proportional  to  the  velocity  the 
particle  will  jjossess  on  reaching  that  point.     Let  us  take  the 


horizontal  and  vertical  lines  through  the  highest  given  point  as 
our  axes,  and  take  the  positive  directions  of  X  and  T  as  the  usual 
negative  directions.  The  velocity  acquired  by  a  particle  sliding 
from  0  to  Q  is,  by  Art.  118,  the  velocity  it  would  acquire  falling 
from  ^to  Q,  that  is,  '^(2gy).  We  shall  have  then,  as  the  de- 
fining property  of  the  required  curve, 


COST 


=  /^, 


where  ^  is  a  constant ;  or  cos  r  =  Oj/^, 

C  being  some  constant.     The  cycloid  is  a  cui^e  possessing  this 
proi:)erty^  as  is  easily  seen. 

Y  T 


•     P 

n 

_2r 

/ 

\ 

y          J 

/         ^ 

\          N. 

y 

/x\ 

^^^ 

_^^ 

T      0 

^ 

T 

Chap.  X.]  INFINITESIMALS.  181 

We  have  cos  r  =  sin PT'  N=  ^-^ ; 

2r 

but,  by  geometry,  FN='^(2ry)  ; 

hence  cosr  =  ^^^  ^^^  =      ( 1.  )  =  ?/5.  q.e.d. 

r/ie  converse^  that  every  curve  possessing  the  property 

cos  r  =  0^5 

is  a  cycloid^  can  be  proved  analytically  by  finding  its  equation, 
as  follows  :  — 

Let  the  required  equation  be 


y=fx. 

We  know  that 

tan  T  =  D^y, 

cos  r  =  — =  Cyh ; 

1  =  C->[14-(Z>.2/)^, 

^  y 

Call 

c-     ' 

V(2«) 

and  assume 

y  =  a  —  acosd. 

JJqX} 

and  we  have 

J      1  —  cos  /? 
a^sin^^*                   2            1  +  cos^ 

{Dgxy          1-QOSff          1— cos6/ 

182  DIFFERENTIAL   CALCULUS.  [Art.  176. 

(D  ^\2_^  o^^sin^^(l-  cos<?)  _  a^(l-  cos^^)  (1-  cos^) 

=  a\l-cosoy, 
DqX=  a(l—  cosO), 
X  =  a/g(l  —  cos 0)  =  aO  —  asmO  -{-C, 
when  aj  =  0,  2/  =  0,  and  ^  =  0; 

hence  (7=0, 

and  our  equations  are     x=aO  —  a  sin  0 
y  =  a  —  a  cos  0 

the  familiar  equations  of  a  cycloid ;  and  the  brachistocJirone  is 
an  inverted  cycloid  with  its  cusp  at  the  higher  of  the  given  points. 


Chap.  XI.] 


DIFFERENTIALS. 


183 


CHAPTER    XL 


DIFFERENTIALS. 


177.  A  DERIVATIVE  has,  in  effect,  been  defined  as  the  limit  of 
the  ratio  of  infinitesimal  increments  of  function  and  variable. 
Consequent!}' ,  in  getting  a  derivative,  we  can  replace  the  incre- 
ment of  the  function  by  any  quantity  differing  from  it  by  an 
infinitesimal  of  a  higher  order. 

For  example  :  in  getting  D^t?^  we  find 

2xAx  differs  from  J(a^)  by  (^ic)^,  which  is  of  the  second  order 
if  we  take  Ax  as  the  principal  infinitesimal,  and  2xJx  may  be 
substituted  for  J(a^)  in  getting  D^x^,  which  then  equals 

limit  [2^1  _  limit   V2x^^2x. 
Ax=0[_  Ax  J      Jx=0'-     -• 

In  our  old  problem  of  getting  the  derivative  of  an  area  we  can 
use  this  same  principle. 

Take  Jx  as  the  principal  infinitesimal,  then  A  A  and  Ay  are  of 
the  first  order,  by  Art.  149.     A  A  differs  from  the  rectangle  yAx 


y 

^ — - 

y 

by 

/ 

^A 

r 

0 

X 

Ax 

by  less  than  the  rectangle  Ax  Ay,  which  is  of  the  second  order,  by 
Art.  147,  Ex.  ;  and  we  have 


184 


DIFFERENTIAL   CALCULUS. 

yJx 

Ax 


[Art.  178. 


D  A=:  limit  ]  i^  |_.  limit 
Ax^0\_Ax]     Jx=0 


Take  the  problem  of  the  derivative  of  an  arc. 


r 

^1 

by 

y 

X 

0 

X           Ar 

Let  Jic  be  the  principal  infinitesimal ;  then  Js  is  of  the  first 
order.  As  differs  from  its  chord  ^1  {AxY  -\-{Ayy  by  an  infinitesi- 
mal of  a  higher  order,  by  Art.  165.     Hence  we  have 


J.    _  limit  rf!f1=  limit 
^-^-Ja;=0|_Ja;J     Ax=0 


■^/{Axy+{Ayr\_  liinit 


Ax 


limit  r    / 


1  + 


178.    In  general, 

r,  .        limit   r/(^  +  Ja;)-/g"|. 


therefore 


Ax 


where  £  is  an  infinitesimal,  by  Art.  7. 

f{x  +  Ax)  —fx  =  P^fx .  Ax  4-  £Ax. 

But  f(x-\-Ax)—fx  is  the  actual  increment  of /aj,  caused  by  the 
increment  Ax  of  x.  eAx  is  of  as  high  an  order  as  the  second,  if 
we  take  Ax  as  our  principal  infinitesimal ;  and  we  get  the  impor- 
tant result  that  D^fx .  Ax  differs  from  the  actual  increment  of  fx 
by  an  infinitesimal  of  a  higher  order,  and  may  consequently  be 
used  in  place  of  Afx  in  any  case  where  we  have  to  deal  with  the 
limit  of  the  ratio  or  of  the  sum  of  such  increments.     This  quau- 


Chap.  XI.]  DIFFERENTIALS.  185 

tity,  D^fx .  Jx  is  called  the  differential  of  fx,  and  is  denoted  by 
d/x,  d  being  a  s3mbol  for  the  word  differential. 
By  the  definition  of  differential, 

dx  =  Dj.  xJx  =  Jx. 

This  definition  may  now  be  restated  as  follows  :  The  differen- 
tial of  the  independent  variable  is  the  actual  increment  of  that 
variable.  The  differential  of  a  function  is  the  derivative  of  the 
function  multiplied  by  the  differential  of  the  independent  variable; 

or  formulating,  dx  =  Ja;, 

dy  =  D,y.dx, 
y  being  a  function  of  x. 

It  is  to  be  noted  that  a  differential  is  an  infinitesimal  and  that 
it  differs  from  an  infinitesimal  increment  by  an  infinitesimal  of  a 
higher  order. 


179.   Since 

dy  =  D^y.  dXy 

l=A.- 

As,  by  Art.  73, 

^'^=i' 

^»^=l 

Consequently,  if  two  quantities  are  so  connected  that  either  is  ia 
function  of  the  other,  the  derivative  of  either  with  respect  to  the 
other  is  the  actual  ratio  of  the  differential  of  the  first  to  the  differ- 
ential of  the  second. 

180.  The  differential  notation  has  the  advantage  over  the 
derivative  notation,  that  it  is  apparently  simpler,  and  that  the 
formulas  in  which  it  is  used  are  more  symmetrical  than  those  in 
which  the  other  notation  is  employed ;  and  although  the  differ- 
ential is  defined  by  the  aid  of  the  derivative,  and  the  formulas 


186  DIFFERENTIAL   CALCULUS.  [Art.  181. 

for  the  differentials  of  functions  are  obtained  from  the  formulas 
for  the  derivatives  of  the  same  functions,  there  is  a  practical 
advantage^  after  the  formulas  have  once  been  obtained,  in  regard- 
ing the  differential  as  the  main  thing^  and  looking  at  the  derivative 
as  the  quotient  of  tivo  differentials. 

181.    By  multiplying  each  of  our  derivative  formulas  b}^  dx^ 

we  get  the  following  set  of  formulas   for  the  differentials  of 

functions. 

da  =  0 ; 

d{ax)      =adx\ 

(^(ic")       =nx''~'^dx\ 


d{\ogx) 

_dx, 

X 

da' 

=  a''\oga.dx; 

de' 

=  e'dx; 

dsina; 

=  cosx.dx; 

dcosa; 

=  —sin  a;,  da;; 

cZtana; 

=  sec^T.  da;; 

detna; 

=  —  csc^T .  dx  ; 

c?seca; 

=  sec  a;  tan  a;,  da;; 

dcscx 

=  —CSC  a;  etna;,  da;; 

d  versa; 

=  sina;.da;; 

dsirr^x 

dx 

V(i-«^)' 

dcos~^x 

dx 

V(i-^)' 

dt&n-^x 

da; 

l  +  x"' 


Chap.  XI.] 

DIFFERENTIALS. 

^opp-l^    _               ^^ 

■  x^(x'-l)' 

rfcse-'T  -            ^'^ 

xV(x^-l)' 

/7,rprq-ir_            ^^ 

V(2^-^')' 

d(u- 

1-  V  +  w  + )  =  du  4-dv^  div  + 

187 


d(uv)  =  udv  4-  vdu  ; 

,?*           vdu  — udv 
d-      = ^ ; 

V  IT 

dA      =  ydx ; 


The  formula  DJy  =  DJy .  D^y 

is  no  longer  necessar}',  as  it  gives  us 

dfy  =  Dyfy .dy  =  -p.dy  =  dfy,         an  identity. 

Examples. 

Work  the  examples  in  Chap.  FV.  hy  the  differential  formulas 
just  given,  remembering  that 

dx 

182.  The  differential  notation  is  especially  convenient  in  deal- 
ing with  problems  in  integration^  and  leads  to  an  entirely  new 
way  of  looking  at  an  integral. 


188  DIFFERENTIAL  CALCULUS.  [Art.  183. 

Let  y  =  ic^, 

and  suppose  that  x  changes  from  the  value  1  to  the  value  5  ;  to 
find  the  whole  change  produced  in  y.  Let  x  change  b}^  succes- 
sive increments,  each  of  which  may  be  called  Ja; ;  then  the  whole 
change  in  y  is  the  sum  of  the  corresponding  increments  of  ?/, 

which  we  will  indicate  by  2  Jy.     The  whole  change  in  y  is  the 

x=l 

actual  sum  of  these  infinitesimal  increments  ;  it  is  then  the  limit 
of  their  sum  as  Jx  is  indefinitely  decreased,  and  each  Jy  decreases 

x=5 

correspondingly  ;  that  is,  it  is  limit  ^  Jy.     But  as  we  are  deal- 

x=l 

ing  with  the  limit  of  a  sum  of  infinitesimals  where  the  limit  is, 
from  the  nature  of  the  case,  finite,  each  term  may  be  replaced 
by  any  infinitesimal  differing  from  it  b}^  an  infinitesimal  of  a 
higher  order  (Art.  162).  Each  Jy  may  then  be  replaced  by  the 
corresponding  dy,  and  we  get  as  the  whole  change  produced  in 

x=5  x=5 

the  value  of?/,      limit  2  d{x^)  =  limit  ^  2xdx. 

x=l  z=l 

As  y  =ar^, 

this  change  must  be 

[x%,-[a;^,.,  =  25-l  =  24, 

and  we  get  the  limit  of  the  sum  of  a  set  of  differentials  appear- 
ing as  the  difference  between  two  values  of  the  corresponding 
function. 

183.  Suppose  that  in  any  fx  we  change  x  from  a^o  to  x^  by 
giving  to  x  successive  increments.  The  whole  change,  fxi  — /Tq, 
must  be  the  sum  of  the  partial  changes  produced  by  the  incre- 
ments given  to  x  ;  or 

fx,-fx,=:s  jfx. 


Chap.  XI.]  DIFFERENTIALS.  189 

If  the  increments  given  to  x  be  indefinitely  decreased  in  magni- 
tude while  sufficiently  increased  in  number  to  still  fill  the  gap 
between  a^o  and  a^j, 

A  -fxo  =  j^^  Q  -2"  J/x  =  limit  ^  dfx, 

X  ^  Xq  X  —  Xq 

by  Arts.  162  and  178, 

=  limit  2  D^fx .  dx. 

Call  DJx  =  Fx, 

then  fx=AFx 

x=x^ 

and  limit  ^  Fa; .  c^x  =  lf,Fx^a;=x,  -  [f.Fx']^^^^, 

X=:X^. 

and  the  limit  of  the  sum  of  a  set  of  differentials  is  the  difference 
between  two  values  of  an  integral.     Such  a  limit  is  called  a  deji- 

x^ 

nite  integral,  and  is  indicated  by  /,  Xq  and  x^  being  the  values 

Xq 

between  which  the  sum  is  taken.  As  a  definite  integral  is  the 
difference  between  two  values  of  an  ordinary  integral,  it  contains 
no  arbitrary  constant. 

184.  Regarding  an  integral  as  the  limit  of  a  sum  gives  a  new 
meaning  to  some  of  our  old  formulas.  Take,  for  example,  the 
case  of  finding  an  area.  Required  the  area  bounded  by  the 
parabola  y^  =  4x,  the  axis  of  X  and  any  ordinate  ?/o- 

The  area  in  question  is  the  limit  of  the  sum  of  rectangles  of 
which  yJx  may  be  taken  as  any  one,  and  the  sum  is  to  be  taken 
between  the  values  0  and  Xq  of  x.     We  have  then 

X  —  Xq  X  ^  X(y 

A  =  limit  2  yJx  =  limit  ^  ydx  ; 

x=0  x=0 

Xo 

hence  A=fydx, 

0 


190 


DIFFERENTIAL   CALCULUS. 


[Art.  186. 


y  =  2xh. 


A  = 


^i^""m„,-[fl...='-'- 


185.  We  can  now  take  up  some  new  problems  that  could  not 
be  conveniently  approached  while  the  integral  was  treated  merely 
as  an  inverse  function,  and  we  shall  consider  very  briefly  one 
connected  with  the  subject  of  centre  of  gravity. 

The  centre  of  gravity  of  a  bod}'  is  a  point  so  situated  that  the 
bod}'  will  remain  motionless  in  any  position  in  which  it  may  be 
placed,  provided  this  point  is  supported. 

Suppose  a  heavy  plane  curve,  of  which  equal  areas  have  equal 
weights,  placed  in  a  horizontal  position.  The  tendency  of  any 
particle  to  produce  rotation  about  a  given  axis  is  the  weight  of 
the  particle  multiplied  by  its  distance  from  the  axis.  If  the  axis 
passes  through  the  centre  of  gravitj',  the  sum  of  all  these  ten- 
dencies must  be  zero,  or  the  bod}'  would  rotate. 

Let  us  consider  the  centre  of  gravity  of  a  segment  of  the  pa- 


rabola 


2/2  =  2mic, 


cut  oflfby  any  double  ordinate. 

Suppose  the  parabola  horizontal,  and  let  X  and  Fbe  the  coor- 
dinates of  the  required  centre  of  gravity.    Inscribe  in  the  parabola 


Chap.  XI.] 


DIFFERENTIALS. 


191 


small  rectangles  having  their  sides  parallel  to  the  axis  of  Y. 
The  tendenc}'  of  any  one  of  these  rectangles,  as  AB^  to  produce 
rotation  about  the  ordinate  through  the  centre  of  gravit}',  is  its 


weight,  which  ma}-  be  represented  b}-  its  area,  2yJx,  multiplied 
by  its  distance  from  the  ordinate  in  question.  If  the  rectangle 
were  so  narrow  that  we  could  reojard  its  weiorht  as  concentrated 
along  its  nearest  side,  this  distance  would  be  (x—X);  and  if  we 
decrease  dx  indefinitely',  the  required  distance  will  approach  this 
as  its  limit. 

The  tendency  of  this  rectangle  to  produce  rotation  is  then, 
roughly,  2y{x—X)Ax;  and  the  smaller  the  value  of  Jic,  the 
nearer  this  comes  to  being  an  exact  expression.     The  tendency 

of  all  the  rectangles  is  ^'2y{x—X)Jx.     The  smaller  the  rect- 

x=0 

angles,  the  nearer  their  sum  comes  to  the  whole  area  of  the  curve, 
and  we  shall  have  as  the  tendency  of  the  whole  curve  to  rotate 

X=X^  X, 

about  CD  lim\t^2y{x—X)Jx  or  f2y{x—X)(lx\  but  as  CD 

X=Q  " 

passes  through  the  centre  of  gravity,  this  must  equal  zero. 

f2y{x-X)(lx  =  0. 
y  =  ^{2mx); 


192  DIFFERENTIAL   CALCULUS.  [Art.  186. 

hence  2/V  (2mx)  {x  —  X)  dx  =  0, 

0 

fxi  dx  =  Xfx^  dx, 

0  0 

ix,i  =  ^Xx,i, 

By  similar  reasoning,  we  find,  as  the  tendency  to  rotate  about 

a  line  through  the  centre  of  gravity  and  parallel  to  the  axis  of  X, 

»i 

f{Xi  —  x)  {y  —  Y)  dy.     This  must  equal  zero. 

-y\  2m       2m/ 


\x^  yl-x^Yy-t^Jr  —  'Y 
[2  ^      ^m  ^  6wi 


'-''    ^0; 


^^y^'  X.  Ym    y'  4-  y^^^  ""^y'  xYv  M  y'  m  y^  y-  o  • 

2x,y^Y=0', 


y.' 


V3m 

r=  0 ; 
and  (^a;i,0)  is  the  required  centre  of  gravit}'. 

Differentials  of  Different  Orders. 

186.  As  the  differential  of  a  function  is  by  definition  a  new 
function  of  the  independent  variable,  we  may  deal  with  its  dif- 
ferential. 

d(dy)  is  called  the  second  diffei-enticd  of?/,  and  is  denoted  by 
d?y ;  d{d^y)  is  called  the  tJiird  differential  of  y,  and  is  denoted 

by  d^y  ;  and  so  on.         d(f?**~^2/)  =  <^"y- 


Chap.  XI.]  DIFFERENTIALS.  193 

In  dealing  with  differentials  of  a  higher  order  than  the  first,  it 
is  customary  to  make  the  assumption  that  the  differential^  that 
is,  the  increment  (Art.  178),  of  the  independent  variable  is  con- 
stant, since  this  assumption  greatl}'  simplifies  the  results,  and  is 
alwaj^s  allowable  when  the  variable  in  question  is  really  inde- 
pendent, as  we  can  then  suppose  it  to  change  by  equal  incre- 
ments. 

187.  Making  the  assumption  that  the  differential  of  the  inde- 
pendent variable  is  constant,  we  have  ver}'  simple  relations  be- 
tween differentials  and  derivatives  of  diflferent  orders. 

By  Art.  1 78,  dy  =  D^y.  dx, 

then   d'^y  =  d{dy)  =D^dy. dx  =  D^{D^y.  dx)dx  =  D^y.  dx^, 

as  dx  is  a  constant.     It  can  be  shown  in  the  same  way  that 

d^y:=D^y.ddi?, 

and  that  d'^y  =  7)/?/.  dx'^. 

It  will  be  noticed  that  when  dx  is  the  principal  infinitesimal, 
d^y  is  an  infinitesimal  of  the  nth  order. 
Fron^the  results  just  obtained,  we  get, 

'^      dx" 


Vy  = 


Dj'y  = 


(Py 
da^' 

d^ 
dx""' 


and  the  differential  notation  is  generally  used  in  place  of  the 
derivative,  even  in  the  case  of  derivatives  of  higher  order  than 

the  first;  but  in  using  _!l  for  7)/?/,  it  must  be  kept  in  mind 

dxf" 
that  the  two  expressions  are  equivalent  only  when  x  is  the  inde- 


194  DIFFERENTIAL   CALCTJLITS.  [Art.  188. 

pendent  variable.  If,  for  example,  x  were  a  function  of  a  third 
variable,  and  were  compelled  to  change  in  some  particular  way, 
we  could  no  longer  assume  that  dx  was  constant,  and  the  differen- 
tial expressions  for  the  derivatives  would  be  much  more  compli- 
cated. 

188.   Let  us  work  out  .the  second  derivative  of  y  ivithout  any 
assumption  as  to  the  value  of  dx. 

dy 
1)2 y_  dD^y  _^  dx      dxd^y  —  dyd!^x 


dx         fix  d^ 


since  ^  is  an  ordinary'  fraction,  and  its  differential  can  be  found 
dx 

by  the  formula  ^u^vdu-udv^ 


(1)    Show  that 


Examples. 


T)3y_  d^yda?  —  dxdyd^x  —  Sdxd^yd^x  -f  Sdyd^x^ 
'^■"  d^' 

(2)    If  ^  =  log^, 

find  d^y^  d^y^  and  d^y^  assuming  that  z  is  the  independent  varia- 
ble, and  again  makihg  no  assumption  concerning  0.  Compare 
3'our  last  results  with  those  obtained  by  letting  z=smx,  and 
taking  x  as  the  independent  variable. 

189.  In  using  differentials  of  higher  order  than  the  first,  if 
the  assumption  is  made  that  the  differential  of  the  independent 
variable  is  constant,  it  is  better  to  indicate  this  b}-  preserving 
the  derivative  form,  even  when  using  the  differential  notation. 
Take,  for  example,  the  formula  for  the  radius  of  curvature  of  a 

plane  curve,  p  =  -  [l±i^,^\ 


Chap.  XI.]  DIFFERENTIALS.  196 


We  should  write  it      p  = 


da? 


and  not  ,=  _(!^±|^', 

dx .  d^y 

if  we  wished  to  indicate  that  x  was  the  independent  variable.  If 
we  make  no  such  assumption,  we  must  substitute  for  D^y  the 
value  given  in  Art.  188,  and  we  can  then  reduce  the  formula  to 

^^        [_dx^-\-df]l 
daxPy  —  dyd^x' 

190.  The  subject  of  differentials  of  different  orders  is  closely 
connected  with  that  of  Jinite  differences  or  increments  of  different 
orders. 

If  2/  is  a  function  of  a?,  and  any  fixed  increment  Ax  is  given 
to  «,  there  will  be  produced  a  corresponding  increment  Ay  in 
the  value  of  y  ;  J?/,  however,  is  not  a  fixed  value,  but  varies  with 
the  value  of  a;  considered.     For  example,  if 

y  =  a?. 

Ay  =  ^x'^Ax  +  ^x{Axy  +  {AxY, 

and  is  obviously  a  function  of  a?,  and  therefore  will  be  changed  by 
changing  x.  The  change  produced  in  Ay  by  giving  x  another 
increment,  Ax^  is  called  the  second  increment  of  i/,  and  is  indi- 
cated by  A^y^  and  is  a  new  function  of  x.  The  increment  of  the 
second  increment  is  the  third  increment  A^y^  and  so  on ;  and  in 

general  J(  J"-\y)  =  J"?/. 

K  y  =  ^, 

A'^y^Qx{Axy-{-Q{Axy. 

The  whole  change  produced  in  a  function  by  giving  several  equal 


196  DIFFERENTIAL   CALCULUS.  .      [Art.  191. 

increments  to  the  variable  can  be  neatly  expressed  in  terms  of 
successive  increments. 

Add  Jx  again,  y  becomes  y  -\-  Ay^  Ay  becomes  Ay  -\-A^y^  and  we 

have  f{x  -f-  2Ax)  =  y-{-  2 Ay  +  A'^y. 

Repeat  the  operation,  y  becomes  2/-|-  Ay^  2  Ay  becomes  2{Ay-\-A^y) , 
A^y  becomes  A^y-{-A^y^  and  we  have 

f(x  +  3Ax)  =  y  +  SAy-\-3A^y  +  A^y. 

In  like  manner, 

f(x-h^Ax)'^y-{-AAy  +  6A'y-^AA^y-^A*y. 


Example. 
Show  that,  if 

/[«  +  (»-l)Ja;]  =  y  +  (»-l)zlj/+('^-^)("~^>  A'y 

(«-l)(n-2)(n-3)    ,, 

3!  y-r       ' 


and  that,  consequently,  the  second  formula  always  holds. 

191.  If  Ja;  is  infinitesimal,  we  have  seen  that  dy  differs  from 
Ay  by  an  infinitesimal  of  a  higher  order,  and  therefore  may  be 
used  instead  of  Ay  in  all  cases  where  we  are  dealing  with  the 
limit  of  a  ratio  or  of  a  sum  of  such  increments.  The  same  rela- 
tion holds  between  d^y  and  A^y,  and  in  general  between  d'^y  and 
J"y,  as  we  can  prove  by  the  aid  of  the  following  lemma. 


Chap.  XI.]  DIFFERENTIALS.  197 

Lemma. 

192.  If  a  function  of  x  contains  besides  x  a  letter  a,  which  is 
independent  of  x,  and  becomes  zero  when  a  is  zero,  no  matter 
what  the  value  of  x,  its  derivative  with  respect  to  x  will  be  zero 
when  a  is  zero. 

For,  since  a,  being  independent  of  ic,  is  treated  as  a  constant 
during  the  operation  of  differentiation,  it  can  make  no  difference 
in  the  result  whether  we  give  it  any  particular  value  before  or 
after  that  operation.  But  if  we  give  a  the  value  zero  before  we 
differentiate,  our  function  by  hypothesis  is  equal  to  zero,  and  is 
therefore  constant,  and  its  derivative  is  zero.    Hence  the  lemma. 

It  follows  that,  if  the  function  is  infinitesimal  when  a  is  inftni- 
tesimal,  whatever  the  value  of  x,  its  derivative  with  respect  to  x 
will  also  be  infinitesimal  when  a  is  infinitesimal. 

As  an  example,  consider  the  function  log(l  +  aa;) ,  which  equals 

zero  when  ci  =  0. 


dlog(l-|-  ax) 


=  0  when  a  =  0. 


dx  1  +  ax 


193.   Let  Jx  be  infinitesimal.     Then,  by  Art.  178, 

Ax 

where  e  approaches  zero  as  Ax  =  0.     Increase  x  by  Ax,  and  the 
increments  of  the  two  members  of  the  equation  will  be  equal. 

^  =  A{D^y)  +  Ae.- 
Ax 


Divide  by  Jx :  - — ^  =     ^  /^^  + 


limit 
Ax 


(AxY  Ax  Ax 

nit   f  ^'y  l_i>2    4_  limit  V^l. 
=  Ol{Axyj~    ''^^Ax=OlAx} 


198  DIFFERENTIAL  CALCULUS.  [Art.  193. 

but,  by  Art.  1 92,  ^^"^^  [—1  =  0. 

(Axy      da^^ 
where  a  is  infinitesimal,  by  Art.  7. 
But  Jx  =  dx  (Art.  178)  ; 

hence  — ^  =  — ^  4-  a, 

J^y=d^y-\-  adoi?. 

cPy  is  an  infinitesimal  of  the  second  order,  by  Art.  187.  ado? 
is  of  the  third  order ;  consequently,  d'^y  may  be  used  in  place  of 
A^y  in  problems  concerning  the  limit  of  a  ratio  or  of  a  sum. 

By  similar  reasoning,  it  can  be  shown  that 

A^y=id^y-\-adQi?; 
and,  in  general,  that  J"?/  =  <^"2/  +  «<^ic'*, 
when  Ax  is  infinitesimal. 


Chap.  XII.]  FUNCTIONS,  ETC.  199 


CHAPTER   XII. 

FUNCTIONS   OF  MORE   THAN   ONE   VARIABLE. 
Partial  Derivatives. 

194.  Up  to  this  time  we  have  considered  only  functions  of  a 
single  variable,  but  a  complete  treatment  of  our  subject  requires 
us  to  study  functions  of  two  or  more  independent  variables. 

Plane  Analytic  Geometry  has  furnished  us  with  numerous  ex- 
amples of  functions  of  the  former  kind ;  Analytic  Geometry  of 
Three  Dimensions  introduces  us  to  functions  of  the  latter  sort. 

The  equation  of  a  surface  contains  three  variables,  «,  2/,  and  2;, 
and  any  one  may  be  expressed  as  a  function  of  the  other  two ; 
and  when  this  is  done,  the  one  so  expressed  may  be  changed  by 
changing  either  of  the  others,  or  by  changing  them  both,  as  they 
are  entirely  independent. 

195.  The  derivative  of  a  function  of  several  variables  obtained 
on  the  h3'pothesis  that  only  one  of  them  changes,  is  called  2i  par- 
tial derivative;  and,  as  all  the  variables  except  one  are,  for  the 
time  being,  treated  as  constants,  2i  partial  derivative  can  be  ob- 
tained by  the  rules  for  differentiating  a  function  of  one  variable. 

For  example  :  D^x^y  =  2^2/,  if  x  alone  changes  ; 

Dy7?y  =  x^.,  if  2/  alone  changes. 

2xy  is  the  partial  derivative  of  x^y  with  respect  to  ic,  and  m?  is 
the  partial  derivative  of  o^y  with  respect  to  y. 

We  shall  represent  partial  derivatives  by  our  old  derivative 
notation,  indicating  ordinary  or  complete  derivatives,  when  it  is 
necessary  to  make  any  distinction  between  the  two,  by  the  ratio 
of  two  differentials. 


200  DIFFER:^TIAL   calculus.  [Art.  196. 

196.  If  a  function  contains  two  variables,  its  partial  deriva- 
tive with  respect  to  either  will  generally  contain  both  variables, 
and  may  be  differentiated  again  with  respect  to  either  of  them. 

Take  ^y^. 

D^^]f=  2x7^ ; 

DyDyy^  =  4:xy\ 
D,x^y'=2x^y; 
D^DyX^7/  =  4.xy\ 
Dy^x?f=2x?. 


Take  u=^x\o^y. 


D,u  =  logy; 
DyD,u  =  l; 

y 
y^ 
y 

D^DyU  =  -; 

y 
y^ 

197.    In  both  these  examples  we  see  that  D^D^u  is  the  same 
as  D^DgU,  and  in  the  second 


Chap.  XII.]  FUNCTIONS,  ETC.  201 

Let  us  see  whether  it  is  true  in  general  that  the  order  in  which 
the  differentiations  are  performed  is  immaterial. 

Let  u=f{x,y). 

To  see  if  Dy D, u  =D,D^u. 


'f{x,y-\-Jy)-~f{x,y) 


f{x,y+Jy)-f{x,y) 

^y 


by  Art.  7,  where  e  is  an  infinitesimal  and  a  function  of  a;,  ?/,  and 

Ay.    Similarly,  D.u  ^/(^±Jm)zL«M)  +  c', 

Jx 

where  e'  is  an  infinitesimal  and  a  function  of  ic,  y,  and  Ja?. 

D^Dyii  is  equal  to 

limit   r/(a;  +  ^a;,7/+^y)  -f{x  +  Ax,y)  -f{x,y-\-Ay)  +/(a;,y)"| 
Ja;=0 1_  Ax  Ay  J 

+A^;  [1] 

DyD^u  is  equal  to 

limit  \A^  +  ^^^y  +  -^y)  -/(-^^y  +  ^y)  -fi^  +  ^a;,y)  +/(a;,y)1 
Ay  =  0\_  Ax  Ay  J 

+  2),e'.  [2] 


The  second  expression  for  DyU  is  absolutely  true,  whatever  the 
value  of  Ay,  and  so  is  the  expression  for  D^DyU.  We  may  then 
suppose  Ay  to  approach  indefinitely  near  zero,  and  D^DyU  will 
be  equal  to  the  limiting  value  approached  by  the  second  member 
of  [1].     The  limit  of  e  as  Ay  approaches  0  is  0  ;  therefore,  by 

Art.  192,  J!,'^oC^'^3  =  0. 

and  D^DyU  is  equal  to 

i:^:.  [f{^  +  ^x,y  +  Ay)  -f{x  4-  ^x,y)  -fjx.y  +  ^y)  -hf(x,y)'\ 
^''^''  I A^^y  J' 

as  both  Ay  and  Ax  approach  0. 


202  DIFFERENTIAL   CALCULUS.  [Art.  198. 

fey  similar  reasoning,  it  may  be  shown  that  DyD^^i  is  this  same 
limit,  and  hence  that     D^DyU=DyD^u. 

By  applying  this  theorem  at  each  step,  we  may  prove  that,  in 
obtaining  any  successive  partial  derivatives,  the  order  in  which  the 
differentiations  occur  is  of  no  consequence. 

For  example,  let  us  show  that 

DJD,u=DyD,hi; 

D,^DyU=D,{D,DyU)=D,{DyD,u)=D,DyD,u 

^DyD^D^u^ByD^u. 

198.  In  a  previous  chapter,  we  saw  that,  while  the  increment 
of  a  function  due  to  any  increment  of  the  variable  is  generally  a 
very  complex  expression,  the  differential,  of  the  function, which 
differs  from  the  true  increment  only  by  an  infinitesimal  of  a 
higher  order  than  the  increment  of  function  or  variable  when 
the  latter  is  infinitesimal,  is  usually  very  much  simpler,  and  yet 
can  be  used  instead  of  the  true  increment  in  many  important 
problems. 

It  is  worth  while  to  see  if  we  cannot  get  a  simple  expression 
capable  of  replacing  the  infinitesimal  increment  of  a  function  of 
two  or  more  variables  in  similar  problems. 

A  function  of  two  independent  variables  maj^  be  changed  by 
changing  either  of  the  variables  alone,  or  by  changing  both. 

Suppose  we  give  to  each  variable  an  infinitesimal  increment 

of  the  same  order.     Let     u=f{x,y). 
Increase  x  by  Jx  and  y  hy  J?/, 

All  =f{x  +  Ax,y  +  Jij)  -f{x,y) . 
Add  and  subtract /(a;, 2/  +  J2/)5  and  we  get 

Ju  =f{x  +  Ax,y  +  Ay)  -f(x,y  +  Ay)  +f{x,y  +  Jy)  -f(x,y) . 


Chap.  XII.]  FUNCTIONS,  ETC.  203 

f(x,y+Jy)—f{x,y)  is  the  increment  of  f{x^y)  produced  by 
changing  y  alone,  and  differs  from  Dyf{x,y)Jy  by  an  infini- 
tesimal of  a  higher  order  than  Jy,  by  Art.  178.  In  like  man- 
ner, we  see  that  f{x-\-Jx^y-\-Jy)—f(x,y-\-Ay)  differs  from 
D^f{x^y-\-dy)Jx  by  an  infinitesimal  of  a  higher  order  than  Jx. 

D^f{x,y  +  Ay)  is  a  new  function  of  x  and  y,  and  any  infinitesi- 
mal change  in  y  will  produce  in  it  a  change  of  the  same  order, 
by  Art.  149.  D^f{x^y -\-Jy)^  then,  differs  from  D^f{x^y)  by  an 
infinitesimal  of  the  same  order  as  dy^  and  D^{x,y -\-Jy)Jx 
differs  from  D^f{x,y)  Jx  by  an  infinitesimal  of  the  second  order. 

^xf{x,y)Ax-j-Dyf{x,y)Jy,  or,  using  the  differential  notation 
and  remembering  that  x  and  y  are  both  independent,  D^f{x,y)dx 
-{-Dyf{x^y)dy  differs  from  the  true  increment  of  w  by  an  infini- 
tesimal of  a  higher  order  than  dx  and  dy^  and  therefore  may  be 
used  in  place  of  Au  whenever  the  limit  of  a  ratio  or  the  limit  of  a 
sum  is  sought.  This  is  called  the  complete  differential  of  it,  and  is 
indicated  by  du  ;  hence,  when 

y'=f{x,y), 

du  =  D^  udx  H-  Dy  udy. 

Example. 
Prove  that,  if  u  =f{x^y,z) , 

du  =  D^  udx  -\-  Dy  udy  +  D^  udz. 

199.  Partial  derivatives  may  very  often  be  used  with  profit  in 
obtaining  ordinary  or  complete  derivatives.     Suppose  that 

y  z=Fx  and  z  =FiX  and  u  =f{y,z) ; 

u  is  indirectly  a  function  of  a;,  and  we  can  therefore  speak  of  the 
complete  derivative  of  u  with  respect  to  a?,  which  we  shall  indi- 
cate by  — 

-^  dx 


204  DIFFERENTIAL  CALCULUS.  [Art.  200. 

We  wish  to  find  the  limit  of  the  ratio  — .     In  so  doing,  we 

Jx 
can  replace  Au  by  du,  which  equals  DyUAy  -\-DgUAz^  since,  as 
y  and  z  are  not  independent  variables,  Ay  and  As  differ  from  dy 
and  dz ; 

hence 


or 


^_  limit 
dx     Ax=0 

[^'"l+^-'f]' 

du      r^     dy      r^     dz 
dx        '    dx^         dx 

Example. 

d8ln(/-.) 

r  2/ =  logic, 

fvinff  that  \ 

^^  [z^x", 

Solution :         Dy  sin  (2/^  —  z)  =  2y  cos  {y^  —  z) 

D^  sin  (2/^  —  z)——  cos  (2/^  —  k) . 


dy 
dx 

1 

dz_ 
dx 

2:k, 

dsin(2/^  — 2)       2yeos(y^  —  z)      j.         /  2       \ 
dx  X  \o        / 

2(y-xP)QOs{y^-z) 


Confirm  this  result  by  expressing  y  and  s  in  terms  of  x  before 
differentiating. 

200.    If  u=f{x,y)  and  y=Fx, 

the  formula  of  the  last  article  becomes 

du  J.  ,  j^  dy 
—  =D,u-\-DyUj.. 
dx  dx 


Chap.  XII.] 


FUNCTIONS,    ETC. 


206 


(1)    u  =  z^-hf-\-^y 
z  =  sinaj 
y  =  e' 


Find 


Examples. 

du 


dx' 


du 


Alls.    —  =  {^y^-\-z)e''  +  {2z-\-y)co&x. 

(XX 


y  =  sin  X  J 


(3)    u  =  tSLn~^{xy) 
y  =  e' 


Find  — . 


A        du      1        . 

Ans.   —  = etna;. 

dx      X 


Ans.    ^^f!^±l. 
dx      1 4-  x^y^ 


(4)  w  =  sin  ^(  _  )  when  z  and  y  are  functions  of  a;.     Find  —  • 

\yj  dx 

(5)  u=^  I    o~    ol  when  2;  and  w  are  functions  of  x.     Find  — . 

\\z'-\-y'J  ^  da; 


201.  Higher  derivatives  of  a  function  of  functions  of  x  can 
be  obtained  by  an  easy  application  of  the  method  suggested  by 
the  formulas  above. 

For  example  ;  u  =f(^y^z) , 


.     -,  (^u 
required . 

do(^ 


d^i 


y=Fx, 

z=F^x, 


0 


dx^         dx 


[^>l+^'^-l] 


dy 
dx 


+ 


[aa.|+z»|]|+[z>,.--,A 


dx^ 


.»(DV2z>„A4:.|-+^/«(l)Vx>,»g+z>.„ 


dsi?' 


206  DIFFERENTIAL   CALCULUS.  [Art.  202. 


In  obtaining  this  formula,  since  y  and  z  are  given  functions  of 

a;,  -^  and  —  are  also  explicit  functions  of  it',  and  are  therefore 
dx         dx 

treated  as  constants  in  obtaining  the  partial  derivatives  with 

du 
respect  to  y  and  z  ;  but  now  —  is  a  function  of  {x^y  and  z) ,  hence 

OjX 

we  must  take  also  its  partial  derivative  with  respect  to  x. 


Example. 
Given  u  =f(x,y) , 


,  .   .    d^u       ,  ^u 
obtain  — -  and  — -. 
doer  doer 


y=^Fx, 


Implicit  Functions. 


202.    If,  instead  of  having  y  given  in  terms  of  x^  we  have  an 
equation  connecting  x  and  y,  y  is  called  an  implicit  function  of 

X,  and  —  can  be  readily  found  by  the  aid  of  Partial  Derivatives. 

dx 

Suppose  /(^,2/)  =  0, 

to  find —.     Call  f(x,y)  u, 
dx 

Then  u  =  0; 

hence  —  must  also  equal  zero, 
dx 

dy  _      D^u 
dx  DyU 


Examples. 


(1)    aa;'»-2/e^''  =  0.     Find  ^. 

dx 


OF   THK 

U.;JIVERSITT 


Chap.  XII.] 

Solution  : 


or,  as 


CaliforH}^ 


FUNCTIONS,    ETC. 

DyU=  —  eT'y  —  nye^\ 

dy  _—D^u  _     max^~'^ 
dx ""    DyU    ~  (l-\-ny)e''" ' 

ax'^  =  ?/e~y, 

mye*^^ 


maaf^~^  = 


X 


and 


my 


dy^ 

dx      {l-\-ny)x 


a^     Ir  dx 

(3)  xy-y'=0.     Find^. 

dx 

(4)  sin(;r?/)  —  mx  —  0.     Find  -^. 

dx 


(5)    u^-^x^  +  y''  +  z'=c' 


\og{xy)  +  -=a' 


lOg(-)  +20^=^,2 


Find 


dM 
dx' 


Ans.   ^=_^. 
dx  o?y 

dx     ar^  —  xylogx' 


Ans.    <i^__Uy\^-y)   I  z\xz-l)      ^ 
dx      u\x{x-\-y)  ^  x(xz-{-l) 


203.   We  can  get  ^  by  the  aid  of  the  formula  of  Art.  201, 
remembering  that  -^  =  0. 

(XX^ 


208  DIFFERENTIAL   CALCULUS.  [Art.  203 

dy  _      D^u 
dx~      DyU 


A^.-2(AZ>,.g)+Z)/.(|-^)Vz),. 


d^     ' 


d'y^      DJ^u{DyUy-2D^DyuD^uD^u-{-D;u{D,u)' 
dx"  {Dyuf 


Examples. 


(1)    f  +  ix?'-2>axy  =  0.   Find  ^.      Ans.   ^  =  - 


2a^xy 


dx^  dx^  (y^  —  ax) 


(2)    x*-^2ax'y-af  =  0.     Find  ^  and  ^. 
^  ^        ^         ^        ^  dx         dx" 


Chap.  XIII.]  CHANGE   OF   VARIABLE.  209 


CHAPTER  XIII. 

CHANGE   OF   VARIABLE. 

204.  If  we  use  the  differential  notation,  we  have  seen  that 
there  is  no  need  of  distinguishing  carefully  between  function  and 
independent  variable,  a  single  formula  alwaj's  giving  a  relation 
between  the  two  differentials  by  which  either  can  be  expressed 
in  terms  of  the  other.  This,  however,  is  the  case  only  when  we 
are  dealing  with  differentials  of  the  first  order.  A  differential 
of  the  second  order  or  of  a  higher  order  has  been  defined  by  the 
aid  of  a  derivative,  which  alwaj's  implies  the  distinction  between 
function  and  variable,  and  on  the  hypothesis  of  an  important 
difference  in  the  natures  of  the  increments  of  function  and  varia- 
ble ;  namel}',  that  the  increment  of  the  independent  variable  is  a 
constant  magnitude,  and  that,  consequently,  its  derivative  and 
differential  are  zero. 

If,  in  any  function  involving  differentials  of  a  higher  order 
than  the  first,  we  have  occasion  to  change  the  independent  va- 
riable, we  can  no  longer  assume  that  the  differential  of  the  old 
independent  variable  is  constant,  but  must  go  back  and  replace 
all  the  differentials  of  higher  order  than  the  first  by  values  ob- 
tained on  the  supposition  that  all  the  differentials  are  variable, 
before  we  attempt  the  introduction  of  the  new  variable,  vide 
Arts.  187  and  188. 

205.  In  an}^  particular  example  in  which  it  is  necessary  to 
change  the  variable,  the  method  just  described  can  be  easily 
applied. 

Take  the  differential  equation, 

od~U    ,  du     ,  ri 

x^  — -  +  a; \-uz=0^ 

dar         dx 


210  DIFFERENTIAL  CALCULUS.  [Art.  205. 

where  x  is  the  independent  variable,  and  introduce  y  in  place  of 

X.     Given  y  =  logo;. 

Our  d^u  here  is  the  second  differential  of  u  taken  on  the  assump- 
tion that  X  is  the  independent  variable,  and  this  can  be  indicated 
by  writing  it  d^u^  and  we  have 

d^ u  =  i)/ uda^  =  dxd^u-dud^x^       .     ^^^^  ^ gg^ 
dx 


,       dx 
dy  =  —, 

X 


dx  =  xdy, 

d^x  =  d{xdy)  =  xd^y-\-  dxdy ; 

but  d22/=0, 

as  y  is  to  be  the  independent  variable, 

hence  d^x=  dxdy, 

^^^  ^,^ ^  xdy<Pu  - xdudf  ^  ^^^ _ 

xdy 


'jd^u  _  d^u      du 
dx^      dy^      dy^ 


du     du 
dx      d.y 


d^u 
hence  we  have  — -  -\-u  =  0. 

dy^ 


Chap.  XIII.]  CHANGE   OF   VARIABLE.  211 

Examples. 
(1)    Change  the  variable  from  a;  to  Hn 

d^y ^     dy  ^       y     =0. 

dx^      l  —  c(?dx      1  —  a^ 


Given  a;  =  cos^. 


Ans.   — ^  _j-  w  =  0. 


(2)    Change  the  variable  from  x  to  6  in  the  equation, 


Given 


d^y 
dx" 

2x     dy 
1-^x^dx 

0  =  tan- 

y    = 

-.0. 

Ans 

d^y 
'    dd' 

-hy  = 

:0. 

[rom 

»  to  Hn . 

<-->g- 

dx 

x=  cost. 

Ans. 

d^y_ 
df 

:0. 

Given 


206.  It  is  often  desirable  to  change  both  variables  simulta- 
neously, and  the  principles  already  explained  and  illustrated 
apply  perfectly'  to  this  case.  As  an  example,  let  us  see  what 
our  old  expression  for  the  radius  of  curvature  of  a  plane  curv^e 
becomes  when  we  change  from  rectangular  to  polar  coordinates. 

Here  we  have  x  =  r  cos  <p 

y  =  rsin^ 

and  we  shall  regard  <p  as  the  new  independent  variable.     We 
know  that,  if />  is  the  radius  of  curvature, 


212  DIFFERENTIAL   CALCULUS.  [AnT.  206. 

,._■     [i+(At/)^i 

We  have  seen,  in  Art.  189,  that  this  may  be  written 

or,  better  still,  (,  =  -    (<i^  +  <¥)i   . 

dxd^y  —  dyd^x 

dx=  —r  sin  <pd(p  +  cos  (fdr^ 
dy  =  r  cos  ifd(p  +  sin  (pdr. 
Since  d(p  is  constant, 

d^a;  =  _  r  cos  (fdf^  —  2  sin  cdrd(p  -\-  cos  ^d^r, 
c?^2/  =  —  ^sin  (pdf'^  +  2  cos  <pdrd(p  +  sin  9?^^?-, 

{dx^  +  d?/2)l  =  (7-2  V  ^  ^^)|^ 

_  {i^dcp'^  +  di^Y^ 

^-      i^dcp^-rd^nah'+2di^d<p' 

divide  numerator  and  denominator  by  c?^^, 


P  = 


[^HiJ 


--S+KI) 


Example. 

Find  the  radius  of  curvature  of  the  circle  r  =  cos  (p. 

Ans.   p  =  ^. 


Chap.  XIII.]  CHANGE  OF  VARIABLE.  213 

207.    A  very  simple  example  of  change  of  variable  is  the  fol- 
lowing.    Obtain  the  value  of  tanr  when  polar  coordinates  are 

used.  tanr  =  _^. 

dx 

2/=  rsin^, 

dy  _    r  cos  ^d(p  -f  sin  (fdr 
dx      —  r  sin  (fd(p  -}-  cos  <pdr' 

A  much  simpler  expression  can  be  obtained  for  the  angle  made 
by  the  tangent  with  the  radius  vector,  which  we  shall  call  e. 


tan.  =  tan(r -  y,)  =  tanr-tany^ 
1-|-  tanr  tan  ^ 

.  .  r  sec  <pd0 

tan  r  —  tan  <p  = ZUl , 

—  r  sin  <pd<p  +  cos  <pdr 

l  +  tanrtan^= sec^rA^__ 

—  r  sin  <fd(p  +  cos  <fdr 

tan£=  —, 
dr 

Examples. 

(1)    Obtain  this  value  for  tans  from  a  figure  by  the  aid  of 
infinitesimals. 


214 

DIFFERENTIAL  CALCULUS.                 [Art.  208. 

(2)    If 

£C=rC0S^  j 
2/  =  rsin^J 

show  that 

x-^yf 

dx       dr 

x^-y      ^^^' 
dx 

and  that 

ds^  =  dx^  ■\- dy^ 

becomes 

ds'  =  di''  +  r^d<p\ 

Prove  this  last  result  from  a  figure. 

(3)    If 

x  =  a(l—cost) 
y  =  a(?i^-|-sin^)  J 

express  --|  in 

terms  of  t.                        Ans 

d-y  __      ?i  cos  ^4-1 
dy?             asin^i 

(4)    Given 

x=a cos <p ' 

n 

^(ir 

y=bsm<P  J 

\c(xj  _ 

^111   fovmc  n'P  tr. 

d'y 
dx' 

^  111   tcl  Ulo  Ul   CP« 

Ans, 

(a^sinV  +  ^^cos^^)' 
ah 

208.  The  subject  of  change  of  variable  can  be  easily  treated, 
by  the  aid  of  the  principles  established  in  Art.  88,  without  in- 
troducin    the  idea  of  differentials. 

D^x 
DzX  D,x 


Chap.  XIII.  J  CHANGE  OF  VARIABLE.  216 

D^xD:'y-D,yD,'x 


"■m 


{D^xY 


^,        D^xDly-D^yDlx 

If  a;  and  y  are  given  in  terras  of  2;,  we  can  calculate  the  values 
of  D^x,  D^y,  DJ^x,  and  i>/v/,  and  substitute  tliera  in  these  form- 
ulas.    Take  Example  (3),  Art.  205. 

x=  cost, 
{l-x^)DJ^y-xD^y  =  0, 

^'^-l\x 
J)  2  „      A  xDfy~D,yD^x 

DtX=  —  sin^ 
Dt^x=  —  cos^, 

smt 
^.2 „      -  smtD.h/  +  costD.y 

J^x  y  =  T—TT- 5 

—  sin-^^ 
1  — a^=  sin^^, 
—  sin  tDf- y -\- cos  tDty      cos^Z),?/ 

«"^^ ^^^¥t +  ^ii^  =  ^' 

D?y=0. 

209.  Suppose  we  have  a  function  of  two  independent  varia- 
bles, and  its  partial  derivatives  with  respect  to  them,  and  wish 


216  DIFFERENTIAL   CALCULUS.  [Art.  209. 

to  introduce,  in  place  of  our  old  variables,  two  others  connnected 
with  them  by  given  relations. 

For  example :  let  2  be  a  function  of  x  and  ?/,  and  let  it  be 
required  to  introduce,  instead  of  x  and  y,  u  and  v,  which  are 
connected  with  x  and  y  by  given  equations.  If  the  equations 
can  be  readily  solved  so  as  to  express  u  in  terms  of  x  and  1/,  and 
0  m  terms  of  x  and  y,  we  may  proceed  as  follows  :  — 

After  the  substitution,  z  is  to  be  an  explicit  function  of  u 
and  V.  Suppose  the  substitution  performed.  As  u  and  v  are 
functions  of  x^  z  is  indirectly  a  function  of  x.  To  get  D^u,  we 
suppose  y  constant,  so  that  x  is  for  the  time  being  the  only  inde- 
pendent variable,  and  we  can  get  D^z,  by  Art.  199,  which  gives  us 

D,z=D^zD,u+D,zD,v 

where  all  the  derivatives  are  partial  derivatives.     In  the  same 

way,  I)yZ=D^zDyU+I)„zDyV. 

D^M,  D^v^  ByU,  and  DyV  are  found  from  the  values  of  u  and  v 
mentioned  above,  and  are  generally  functions  of  x  and  y,  and 
D^z  and  D^z  are  at  first  obtained  in  terms  of  i«,  v,  ic,  and  y. 
X  and  y  must  be  replaced  by  u  and  v  by  the  aid  of  the  given 
equations,  and  D^z  and  D^z  are  then  in  terms  of  u  and  v  alone. 
By  extending  the  process,  we  can  get  BJ^z,  D^DyZ,  B/z^  &c., 
in  terms  of  u  and  v. 

For  example :  introduce  u  and  v  in  place  of  x  and  y  in  the 

equation  DJ^ z  =  Dy  z . 

Given  u  —  x-j-y 

v  =  x  —  y 


Chap.  XIII.]  CHANGE  OF   VARIABLE.  217 

DyZ=I)^z  —  D„z, 

D,'z  =  DJz  +  2D,D,z-^DJ'z, 

DJ'z  =  DJz-2D^D„z-^D„H, 

DJz+2D,D„z  H-  D^'z  =  DJz-2D,D„z  +  DJ'z, 

4:B^D,z=0. 

D^D^z  =  0,        the  required  equation. 

210.  If  it  is  more  convenient  to  express  x  and  y  in  terms  of 
II  and  V  at  the  start,  we  can  proceed  thus ;  z  is  explicitly  a  func- 
tion of  X  and  y,  and  if  we  regard  v  as  constant  for  the  time 
being,  z  is  indirectly  a  function  of  the  smgle  variable  u.    Hence, 

D^z  =  D.zD^x  +  DyZD^y  ; 

In  like  manner,       D^z  =  D^z D^x -\-  D^z D„y. 

B^x^  D^y,  D„x,  and  D^y  are  found  in  terms  of  21  and  v,  and 
then  by  elimination  between  the  equations,  we  get  B^z  and  ByZ 
in  terms  of  u  and  v. 

Examples. 

( 1 )    Given  x  =  r  cos  <p 

y  =  rsm<p 

find  B^z  and  B^z  in  terms  of  r  and  <p. 
Solut  ion :  BfX  =  cos  <p , 

Bcpx=  —  rsiny?, 
Bry  =  sin<p, 


218  DIFFERENTIAL    CALCULUS.  [Art.  2U. 

D^z  —  D^zGOSip  +  Dy^sh\<p^ 

D^z=  —  D^zr  sin<p  +  D^z7'cos<p. 
Eliminate ; 

rcos(pD^^=  r cos^ <p D^z -{-  r sin <p cos <pDyZ, 
sm<p  D^z=  —  rsin^<pD^z+  r sin (p  cos <p  D^z  ■ 
D^z=  -  {rcos<pDyZ  —  sin^Z>^2!) 

DyZz=-  {r sm  If  DrZ-{-  cos (pD^z). 
r 

(2)    Solve  this  same  example  by  the  method  of  Art.  209,  usmg 

the  relations  i^  =  x-  +  i/  1 


tan^=  _ 


211,.  If  it  is  not  convenient  to  solve  the  given  equations  be- 
tween a;,  y,  u,  and  y,  we  can  use  the  general  method  of  either 
of  the  precedmg  articles,  obtaining  our  D^u,  D^v,  DyU,  and  D^v, 
or  our  D^x,  DuVt  D^x^  D„y,  as  follows  i   We  have  given 

Fi{x,y,u^v)  =  0  and  F2{x^y^u,v)  =  0. 
Suppose  y  constant,  then  u  and  v  will  be  functions  of  x ;  and, 
by  Art.  200,  D,F,  + D^F^D^u-^  D,F,D/c=:0\ 

D,F,-\-D^F,D^u  +  D,F,D^v  =  o\ 

From  these  equations  we  can  obtain  D^u  and  D^v^  and  from  two 
equations  formed  in  the  same  way  we  can  get  DyU  and  D^v; 
and  a  like  process  would  give  us  X>M^i  J^ul/t  J^v^i  I^vV- 


i 


Chap.  XIIL]  CHANGE   OF   VARIABLE.  219 

Examples. 

(1)  If  F  is  a  function  of  v,  and 

-<;2  =  a;2  4-  f, 

show  that  />/F+  />/ F=  ^^^+  i  ^. 

qW      V  do 

(2)  If  F  is  a  function  of  v,  and 

<;2  =  a;-  4-  //^  +  2;^, 

show  that         DJ"  V+  Dj"  F+  />/  F=  ^4-  ?^. 

(3)  If  a;=rte^cos^    and    y  =  ae^sin<p, 

show  that  /Z>,2  u  -  2xy  D,  D^  u  +  x^  Z>/ 1«  =  Z)^^t  -}-  D^  m. 

(4)  Given  e*  +  e''  =  s, 

express  D^u  +  2D^DyU  -f  Z>/?«  in  terms  of  s  and  ^. 

Ans,   ^D,hi  -2stD,D,u  4-  ^'Z>/^f  +  sD,u-}-  tD^u. 


220  DIFFERENTIAL  CALCULUS.  [Art.  212. 


CHAPTER  XIV. 

TANGENT   LINES   AND   PLANES. 

212.  It  is  shown  in  Analytic  Geometry  of  Three  Dimensions, 
that  any  equation  F(x,y,z)  =  0  represents  a  surface, 
and  that  two  such  equations, 

F2{x,y,z)  =  0, 

regarded  as  simultaneous  equations,  represent  a  curve  in  space, 
the  intersection  of  the  surfaces  which  the  equations  separately 
represent. 

By  eliminating  z  between  these  two  equations,  we  can  express 
y  as  an  explicit  function  of  x ;  and  by  eliminating  ?/,  we  can 
express  z  in  terms  of  x:  consequently,  the  equations  of  any 
curve  in  space  may  be  written  in  the  form, 

z  =  Fx 

213.  Let  it  be  required  to  find  the  direction  of  the  tangent 
line  drawn  at  any  given  point  (iCo?2/o?^o)  of  the  curve 

z  =  Fx 

Let  {xq  +  Ja;,  y^  +  Ay^  Zq  +  J2;)  be  any  second  point  on  the  given 
curve.     The  equations  of  the  line  joining  the  two  points  are 


Chap.  XIV.]         TANGENT  LINES   AND   PLANES. 


221 


~         ""jT' 


J 

X 

^y 

by  Analytic 

Geometry ; 

or 

y- 

-2/o_ 

Ay 

X- 

-^0 

Ax 

z- 

X- 

Az 
Tx 

Let  Ax  approach  zero,  and  the  secant  line  approaches  the  re- 
quired tangent  as  its  limit,  and  this  will  have  for  its  equations. 


y- 

-yo 

fdy' 

X- 

-Xo 

\dx^ 

z- 

-2^0 

fdz' 

X- 

-Xn 

\dx 

I 


:X(,J 


or,  writing  them  in  a  more  symmetrical  form, 

x  —  XQ_y  —  yo    ^  —  ^0 


d]h 
(Ixq 


dzp 
dxo 


where,  by  -^,  we  mean  the  value  —  has  when  x  =  Xq. 
dxQ  dx 

A  plane  through  the  given  point  perpendicular  to  the  tangent 
line  is  called  the  normal  plane  at  the  point  in  question.  Prove 
that  its  equation  is 


(-^h 


dzc 


x-x,  +  {y-yo)'^  +  {z-Zo)':p  =  0 


dxt 


dxQ 


Example. 

214.   The  helix  is  a  curve  traced  on  the  surface  of  a  cylinder 
of  revolution  by  a  point  revolving  about  the  axis  of  the  cylinder 


222 


DIFFERENTIAL   CALCULUS. 


[Art.  214. 


at  a  uniform  rate,  and  at  the  same  time  advancing  with  a  uni- 
form velocity  in  the  direction  of  the  axis. 

We  can  easily  express  its  equations  by  the  aid  of  an  auxiliary 


angle,  the  angle  through  which  the  point  has  rotated.     Calling 
this  angle  d  and  the  radius  of  the  circle  a,  we  readily  see  that 

y=  asin^. 
From  the  nature  of  the  helix,  z  must  be  proportional  to  the 

angle  0  ;  hence  -  =  ^^  a  constant, 


^-Z- 
-_^, 


and  z  =  ko. 

The  required  equations  are  then 

a;=  acos^ 
y  =  a  sin  <? 
z^kd 


Chap.  XIV.]        TANGENT   LINES   AND   PLANES.  223 

To  find  the  tangent  line  and  normal  plane  at  {xQ^yQ^Zg) , 
dy  =  a  cos  Odd, 
dx—  —  a  sin  ddd^ 


^=_ctn^y  = 
dx 

X 

~7 

dz  =  Md, 

dz              k 

dx          a  sin  0 

y 

The  equations  of  the  tangent  are 

x  —  xq  _  y  —  yo  __  ^  —  ^0 

1  Xq  k 

2/0  2/0 

-2/o  ^0  fc    * 

The  normal  plane  is 

2/o(^  -  a^o)  -  My  -  2/o)  -^(2;  -  ^0)  =  0.  [2] 

The  direction  cosines  of  line  [1]  are,  by  Analytic  Geometry, 

-2/0 


cos  a  = 


or  cos  a 


_     -2/0 


cos/?=         /         , 


'^^''=7^+^' 


224 


DIFFERENTIAL   CALCULUS. 


[Art.  215. 


Cos^  is,  then,  not  dependent  on  the  position  of  the  point  P; 
therefore  the  helix  has  everywhere  the  same  inclination  to  the 
axis  of  the  cylinder;  or,  in  other  words,  it  crosses  all  the  ele- 
ments of  the  cylindrical  surface  at  the  same  angle.  If,  then, 
this  surface  is  unrolled  into  a  plane  surface,  the  helix  will  de- 
velop into  a  straight  line. 

215.    The  equations  of  the  tangent  line  to  the  curve 

/(x,?/,^)  =  0, 

F{x,y,z)  =  Q, 

can  be  obtained  in  a  very  convenient  form  if  we  use  partial 
derivatives.     We  have,  by  Art.  199, 


dx  dx  dx 

dx  dx  dx 


(1) 


flfj  (J'y 

From  these  equations  we  can  obtain  the  values  of  —  and  —- 

dx  dx 

Substituting  these  in  the  equations  of  Art.  213,  and  reducing, 
we  get 

{X  -  x,)D^J-\-  {y  -  yo)DyJ+  {z  -  z,)D,J=  0    ] 


(x  -  x,)D^F+  (y  -  yo)DyF-j-  (z  -  z,)  D^^F=  0 


as  the  equations  of  the  required  tangent.     The  same  result  may 
be  obtained  much  more  easily  by  substituting  in  (1)  the  values 

of-^  and  -^  given  by  the  equations  in  Art.  213. 


Chap.  XIV.]       TANGENT  LINES   AND   PLANES.  225 

Examples. 

(1)  Given  x^-\-y^—ax  =  0 

ax -\- z^  —  a^  =  Q 
as  the  equations  of  a  curve,  find  the  tangent  at  (iCoi2/o52;o)  • 

Ans.   XqX  +  y^y  +  ZqZ  =  a^ 

a{x  —  XQ)-^2{z  —  ZQ)zQ=:0 

(2)  Given  the  circle 

a?  +  y^-\-z^=-ir 
x-\-z  =  a 

find  the  tangent  at  (a^oi^/oi^^o)  •  Ans.   XqX  -{-  yoy  -]-  ZqZ  =  a^ ) 

x-\-z  =  a  J 

216.  The  osculating  plane  at  a  given  point  of  a  curve  in  space 
is  the  limiting  position  approached  by  a  plane  through  the  point 
and  two  other  points  of  the  curve  as  the  latter  approach  indefi- 
nitely near  the  given  point. 

If  (xQ.yo^Zo)  is  the  given  point,  and  we  regard  x  as  our  inde- 
pendent variable,  we  can  represent  two  other  points  of  the  curve 
(Art.  190)  by 

{xo  +  Jx,yo  +  ^y,z^-^Jz) 

and  {Xn  +  2Jx,  ?/o  +  2  Jy  +  ^^y,  Zo-\-2Jz-^  A^z) . 

Forming  the  equation  of  the  plane  through  these  three  points, 
dividing  b}'  Ja;^,  and  taking  the  hmiting  values  as  Jx  approaches 
zero,  we  shall  get  as  the  osculating  plane, 

Example. 
Obtain  the  osculating  plane  of  the  helix  at  {xQ^y^^z^) . 


226  DIFFERENTIAL   CALCULUS.  [Art.  217 

217,    The  tangent  plane  at  a  given  point  of  any  surface 
f{x,y,z)  =  0 

can  be  found  by  tlie  aid  of  the  equations  of  Art.  215. 

Let  F{x,y,z)  =  0 

be  any  second  surface  whatever  passing  through  (a^o^^/o^^^o)  • 

The  tangent  hue  to  the  curve  of  intersection  of  the  two 
surfaces  at  the  point  (xo^yQ^Zo),  that  is,  to  any  curve  through 
Cxq^i/q^Zo)  traced  on  the  given  surface,  has  for  its  equations 

(X -  x,)D^/+  (y  -  yo)D,J+  (z  -  z,)D,J^  0 

{X  -  Xo)D^^F+  (y  -  y,)DyF+  {z  -  z,)D,F=  0 

It  therefore  lies  in  the  plane  represented  by  the  first  of  these 
equations,  which  must  then  be  the  required  tangent  plane, 

(x  -  x,)D^J+  (y  -  y,)DyJ+  {z  -  z,)D,J=  0. 

Examples. 

(1)  Find  the  tangent  plane  to  a  sphere. 

x^  +  y'^  +  z^=  d^. 

Ans.    XoX+yoy  +  ZoZ=^a^. 

(2)  Find  the  tangent  plane  to  an  ellipsoid. 


E!-i_^'4--=l 


Ans.    ^,yoy,Zj^ 

^2  t-  52  -f  ^      1. 


The  normal  line  at  (a^o^^/o^^o)  is  easily  seen  to  be 

x  —  Xq  ^  y  —  yo  ^  g  — ^0 
^-0/      ^yof      ^^J 


Chap.  XV.]      FUNCTION   OF   SEVERAL   VARIABLES.  227 


CHAPTER   XV. 

DEVELOPMENT   OF  A   FUNCTION   OF   SEVERAL   VARIABLES. 

218.  To  develop  f{x  -\-h,y  -\-  k)  into  a  series  arranged  accord- 
ing to  the  powers  of  h  and  /c,  where  h  and  k  are  any  arbitrary 
increments  that  may  be  given.     Let  a  be  any  variable,  and  call 

—  =  "ij     —  =  M? 
a  a 

so  that  h  =  ahi  and  k  =  aki. 

If  now  X  and  y  are  regarded  as  given  values,  f{x  +  7i,2/  -f-  k')  is  a 
function  of /i  and  k,  which  depend  on  a  ;  and  hence  f{x  -\-  7i,y  -\-  k) 
can  be  considered  a  function  of  a.  Call  it  Fa^  and  it  may  be 
developed  by  Maclaurin's  Theorem,  which  gives 


Fa=FO-^aF'0  +  —^F"0  +  —^  F"'0  + 

^n\  (^+1)! 

When  a  =  0, 

Fa  or  F(x  -\-h,y  +  k)  =f{x,y) . 
Call  x-{-  ahi  =  x'  and  y  +  «^^i=  2/'? 

then  Fa^f(x\y'), 

aa  da  da 

by  Art.  199  ; 


DIFFERENTIAL   CALCULUS.  CArt.  218. 


da 


da 

F'a  =  h,D^.f{x',y')  +  k,D,.f(x',y'), 
F'O  =  KDJ{x,y)  +  \DJ{x,y) , 
which  we  shall  write     liiD^f-\-  h^Dyf. 

F'^a  =  ^^  =  1i^D,.FU  +  \Dy.  F'a 
da 

=  h^D/f{x\ y')  +  2h,k, D^  Dy, f{x\ y')  +  h'D/f(x   v') , 

F"'a^h,'D/f{^\y')  +  dh,'k,DJD,.f(x',y') 

+  Sh.h'D^.  D/f{x\  y')  +  h'D/f{x',  y') . 

In  F^^a  and  F^'^a  the  terms  have  a  striking  resemblr\ice  to 
the  terms  of  the  second  and  third  powers  of  a  binomia'  >  Let 
us  see  whether  this  will  hold  for  higher  derivatives.  J  ssume 
that  it  holds  for  the  F^^'^a,  and  see  if  it  holds  for  the  F^'"'  ^a. 

If        F^-^a  =  KD,,^f{x',y')  ■^nlil-^\D^^-'Dy,f{x',y') 
^n_{n-})  n,--2j,^'D,--^D/f{x\y') 

F^'^+i)  a  =  /ii  D^,  2^("> «  +  ^1  Dy.  F^''^  a 
+  i^±^'  h--'h'D^.^-'D/f(x',y')  + 


Chap.  XV.]     FUNCTION   OF   SEVERAL   VARIABLES.  229 

If,  then,  the  observed  analogy  holds  for  an}-  derivative,  it  holds 
for  the  next  higher.  It  does  hold  for  the  third ;  it  holds  then 
for  the  fourth,  and  for  all  succeeding  ones. 

n{ii—\) 


+  (n+l)h{'k,D;'DJ(x  +  eh,y  +  Ok)  +  .... 


B}'  this  notation  we  mean  that  x-{-  Oh,  y  -\-  Ok,  are  to  be  sub- 
stituted for  X  and  y  after  the  differentiations  are  performed. 
We  have  then,  remembering  that 

ahi  =  h  and  aki  =  k, 
f{x  +  h,y  +  k)  =f{x,y)  +  {hDJ+  kDJ) 

+  Yi  ('^'AV+  2hkD^DJ+  kWJ^f) 

+  i  {hW^V+  Sh'kDJ^DJ+  Uk^D^DJ'f-h  k^D.'f)  + 

^l-(h-D,-f+nh--''kD:^-''DJ+'^^''- ^^  h^-'lc'D:^-'D^'f-\- ^ 

+  ,     I,,,  (h-^'Dj^^Vi^  +  ^^^y  +  Ok) 

+  {n+l)h''kDj'D^f{x  +  Oh,y  +  Ok) 

+  (!?i±ll2^  h--^T^D:^-'D,'f{x  +  Oh,y  +  Ok)  + Y 

If  we   use    (JiD^-\-kDy)''f  as    an    abbreviation    for   {h'^D^f 
A.nh''-^kDj'-^DJ+ );  that  is,  understanding  that  (/iD.+AjDyV 


230  DIFFERENTIAL   CALCULUS.  [Art.  218. 

is  to  be  expanded  just  as  though  it  were  a  binomial,  and  then  to 
have  each  term  written  before  /,  we  can  simplify  the  above  ex- 
pression. 

f{x  +  h,y  +  h)  =f{x,y)  +  (hD^  +  kD,)f{x,y) 

+  ^  (hD^+kD,yf(x,y)-}-j^^  {hD^  +  kD„yf(x,y)-^ 

+  l.(hD^-\-7cD,rf{x,y) 
n  I 

+  (,,^1);  (^'^^  +  kD^r  +  'fix  +  0h,y -{-Ok), 

which  is  Taylor's  Theorem  for  two  independent  variables. 

If  we  let  x  =  0  and  ?/  =  0, 

we  get  /{h,k)  =/(0,0)  +  (JiD^  +  kD,)f (0,0) 

+  l-(hD^  +  JcD,ffi^,0)-\- ; 

or,  changing  h  and  A;  to  a;  and  ?/, 

fi^.y)  =/(0,0)  +  {xD^  +  yD,)f (0,0) 
+  i^  (xD^  +  yD,rf(0,0)-hj^(xD^-{-yD,yf {0,0)  + 

+  i-  {xD^  +  2//>,)  V(0,0)  +       ^        {xD,  +  yD,)-^'f{0x,0y) , 
n!  (n  +  1)! 

which  is  Maclaurin's  Theorem  for  two  variables. 

Example. 
Transform  Ax"^  +  Bxy  -\- Cy^-  -\-  Dx  +  Ey  +  F  =  0 
to  {xQ^y^)  as  a  new  origin,  the  formulas  for  transformation  being 
x  =  XQ  +  x\y  =  yfi  +  y\ 


Chap.  XV.]    FUNCTION   OF   SEVERAL   VARIABLES.  231 

Call  our  given  equation /(a;, 2/) ;  we  wish  to  develop 

/(a^o-f  a;',  ?/o +  ?/'). 

/(a^o  +  x',  vo  +  y')  =  A^o^yo)  +  ('^'^ar,+  y'Dy^)f{xo,yo) 

H-  ij  {x'D.^-h  y'D,ff{xo.y,)  + 

nj(x,y)  =  2Ax-{-By  +  D, 

DJ(x,y)=^.Bx-{-2Cy  +  E, 

DJ^f{x,y)=2A, 

D^DJ{x,y)  =  B, 

Dlf{x,y)  =  2C; 

all  higher  derivatives  are  0. 

+  {2Ax,  +  By,  +  D)x^+  {Bx,  +  2C2/0  +  ^)y 

+  Ax^^  -\-  Bx'y'-\-  Cy'^,        a  familiar  result. 

219.  By  like  reasoning,  Taj'lor's  Theorem  can  be  extended  to 
functions  of  more  than  two  variables.  For  three  variables  it 
becomes 

f{x  +  h,y  +  k,z  +  I)  =f{x,y,z)  -f  {hD,  +  A'i),  +  lD,)f{x,y,z) 

-h  ^  {liD^  4-  kD,  +  W,yf(x,y,z) 

+  i  (/ii),  +  fcD,  +  W,rf(x,y,z)  +  ..... 

+  7-4-TTT  (^^A  +  ^•I>,  +  W,Y^'f{x  +  ^7i,2/  +  ^^,0  +  dl). 


232  DIFFERENTIAL  CALCULUSc  [Art.  220. 

Example. 

Transform    x^ -{-y^ -\~z^  -  4x-\- ey -2z -11=  0 
to  the  new  origin  (2,-3, l)o  Ans.   x^ -\-y^ -\-z^=2b. 

Elder's  Theorem  for  Homogeneous  FtmctionSo 

220.  A  homogeneous  function  of  several  variables  is  one  of 
such  a  nature  that,  if  each  variable  be  multiplied  b}'  some  con- 
stant, the  function  is  multiplied  by  a  power  of  that  constant. 
The  order  of  the  function  is  the  power  of  the  constant  by  which 
it  is  multiplied. 

For  example  :  oc^ -{- xy  —  y^  is  homogeneous  of  the  second  or- 
der ;  for,  if  we  change  x  into  kx  and  y  into  ky^  our  function 
becomes  ¥{x^  -\-xy  —  y^) ,  and  is  multiplied  by  the  second  power 

of  k.     Sin is  homogeneous  of  the  zero  order ;  for,  if  we 

2x 

multiply  X  and  y  by  A;,  the  function  is  unchanged  ;  that  is,  it  is 
multiplied  by  k^. 

Let/(x,2/)  be  a  homogeneous  function  of  x  and  y\  then,  no 
matter  what  the  value  of  g, 

f{x  +qx,y  +  qy)  =f{x,y)  +  q(xD,  +  yD^)f{x,y) 
+  ^^{xD^-\-yD,Yf{x,y)  + 

+  lmJ^\)\  ^''^^  +  yDyT^'fix  +  qOx.y  +  qOy) ; 
hvLtf{x  +  qx,y  +  qy)=f\_{l-}-q)x.{l-{-q)y'\={l-^qrf{x,y) 
by  the  definition  of  a  homogeneous  function. 

Call  f(x,y)  =  u^  and  we  have 

{l  +  qYu  =  u-\-q{xD,  +  yDy)uJ^t.^{xD,-\-yDyyu 

+  t.{xD^  +  yD,yu  + 


Chap.  XV.]     FUNCTION  OF   SEVEKAL  VARIABLES.  233 

As  this  equation  must  hold,  no  matter  what  the  value  of  9, 
the  coefficients  of  like  powers  of  q  in  the  two  members  of  the 
equation  must  be  the  same.     Equating  them,  we  have 

{xD^-\-yD'y)u  =  nu, 

{xD^-\-yDyyu  =  n{n  —  \)u, 

{xD,  +  yDyYiL=n{n-l){n-2)u, 

(xD^-\- yDy)'^u  =  n(n  —  1)  (n  —  2) (n  —  m  +  l)u; 

and  these  equations  are  Euler's  Theorem. 

Examples. 
Verif;y  Euler's  Theorem  for  second  and  third  derivatives  when 

u  =  Qi^  +  y^  and  when  u  =  sin~^  -. 

X 


234  DIFFERENTIAL   CALCULUS.  TArt.  221. 


CHAPTER   XVI. 

MAXIMA  AND   MINIMA   OF   FUNCTIONS   OF   TWO    OR    MORE 
VARIABLES. 

221.  If  we  have  a  function  of  two  variables  u=f(x,y),  and 
/{xo4-7i,?/o  +  A;)— /(iCoi2/o)<0  for  small  values  of  h  and  k,  ho 
matter  what  the  signs  and  relative  magnitudes  of  these  values, 
w  is  a  maximum  for  the  values  Xq^Pq  of  x  and  y.  lff{xQ  -}-  /i,?/o  -f-  k) 
— /(xo2/o)>0  under  these  same  circumstances,  u  is  a  minimum. 
By  Taylor's  Theorem, 

+  ij  {hD,  +  kD^Yfix,  +  Oh,y,  +  ^A") . 

If  we  take  the  values  of  h  and  k  sufficiently  small,  we  can  always 
make  ^  {JiD,  +  kD^Yf^x,  +  dh^y^  +  Ok) <  (/^D,  +  kBy)  f{x, , 2/0) , 
and  then  the  sign  of  the  second  member  will  be  the  sign  of 
{hD^-{-kDy)f{xQ,yo);  that  is,  of  hDx^UQ-\-kI)y  Uq,  which  evi- 
dently depends  upon  the  signs  of  h  and  k.  In  order,  then,  that 
the  sign  of /{Xq  +  h,yQ  +  k)  —/{xQ^yo)  should  be  constant,  — that 
is,  in  order  that  for  a^o^^/o  ^  should  be  either  a  maximum  or  a 
minimum,  — the  terms  JiD^ii  +  kDyii  must  disappear,  no  matter 
what  the  values  of  h  and  k ;  or,  in  other  words,  D^  Uq  and  Dy  Uq 
must  both  equal  0.  We  get,  then,  as  essential  to  the  existence 
of  either  a  maximum  or  a  minimum,  the  conditions 

D^u  =  0, 

DyU  =  0, 

for  the  values  of  x  and  y  in  question. 


Chap.  XVI.]    MAXIMA  AND   MINIMA   OF  FUNCTIONS.         235 

222.    Carrjing  the  development  a  step  farther,  and  assuming 
that  Dx  Wo  and  Dy  Vq  are  zero, 

f(xo  +  Jhyo  +  k)  -f(xo,yo)  =  —  (hD,  +  TiDyY /{x^.y^) 
+  ^  {hD,  +  kD^y/ixo  +  Oh,yo  +  Ok) . 

As  before,  it  is  evident  that  for  small  values  of  /i  and  A:,  the 
sign  of  the  whole  second  member  will  be  that  of  the  terms 
i  {Ii^Dx^Uq  +  2hkDx  Dy  Uq  +  k^Dy^Uo) .  Let  us  investigate  this 
carefully. 

Let  A  =  Dx^Uq, 

B=Dx^DyUQ, 

C=Dyluo, 

our  parenthesis  becomes  Ah^  +  2Bhk  +  CJi^  ;  and  for  a  maximum 
or  minimum  the  sign  of  this  must  be  independent  of  the  signs 
and  values  of  h  and  k. 

Ah'  +  2Bhk  +  Ck'  =  i  (AVi'  4-  2ABhk  +  ACk^) , 

=  -  {AVi"  +  2ABhk  -^B^T^-  ^k""  +  ACk') , 
A 

=.ll{Ah-\-Bky-{-{AC-B')k'-], 

(Ah  +  Bky  and  F  are  necessarily  positive.  If  AC—B'  is  also 
positive,  the  sign  of  the  whole  expression  will  be  independent 
of  h  and  k,  and  will  be  positive  if  A  is  positive,  and  negative  if 

A  is  negative.     If  AC—  B'=0,     the  result  is  the  same  ; 

but  if  AC—  B'  is  negative,  the  sign  of  the  parenthesis  will  de- 
pend upon  the  sign  and  relative  values  of  h  and  k,  and  we  shall 
have  neither  a  maximum  nor  a  minimum. 


OP  THE 

INTIVERSITY 


236 


DIFFERENTIAL   CALCULUS. 


[Art.  223. 


223.    To  sum  up:  — 

If  D:,^Uo=0 

Da^luoDyluo-  {Da^^Dy^UoY  =  or>0 
If  D^  ?/o  =  0 


Dy^Uo=0 


D,lu,D,fn,-{D,Dy^u,y=  or>0 


DxIuq>^ 


\  Hq  is  a  maximum. 


>  Un  IS  a  minimum. 


Examples. 

224.  (1)  To  find  a  point  so  situated  that  the  sum  of  the 
squares  of  its  distances  from  the  three  vertices  of  a  given  tri- 
angle shall  be  a  minimum.  Let  (a;i,2/i),  (3^2^2/2)5  (^312/3)  ^^  the 
given  vertices,  and  {x^y)  the  required  point. 

u  =  {x-x;)''  +  {y-y^y  +  {x-  x^) ^-\-{y-y,y 

+  (x-x,y-{.(y-y,y 

is  the  function  which  we  must  make  a  minimum. 

D.^uz=2{x  —  Xi)  -\- 2 (x  —  X2) -\-  2(x  —  Xq)  , 

D^u  =  2(y-y,)-{-2(y-y,)-^2(y-y,), 

D^hji  :z=2  +  2  +  2  =  6  =  ^, 

Z>/?^  =  2  +  2  4-2  =  6=(7. 
We  must  make  D^u  and  DyU  both  equal  to  zero. 


Chap.  XVI.]    MAXIMA   AND   MINIMA  OF   FUNCTIONS.  237 

2{x  —  a^i)  -j-  2(a;  —  0^2)  -{-2{x  —  x^)  =  0, 

jp ^1  ~f~  ^2  ~t~  ^3 

~  3 

2(2/ -  2/0  4- 2(2/ -  2/2)  +  2(2/ -  2/3)  =  0, 

,,         ^1  +  ^2  +  2/3 

y  = 3 , 

^e-i32  =  36-0>0, 
^  =  6>0. 
Hence  i*  is  a  minimum  when 

y_^i  +  ^2  +  a^3  oii(i  y_yi±yi±y_3 
~       3  ^~       3      * 

The  required  point  is  the  centre  of"  gravity  of  the  triangle. 

(2)    To  inscribe  in  a  circle  a  triangle  of  maximum  perimeter. 
Join  the  centre  with  each  vertex  and  with  the  middle  point  of 


each  side.  The  angles  between  the  three  radii  are  bisected  by 
the  lines  drawn  to  the  middle  points  of  the  sides.  Call  these 
half-angles  6^1,  O2,  0^. 


-  -1-  r  =  sin  ^1, 
2  ' 


a=  2?'sin^i, 
6=  2rsin^25 
c=  2rsin% 


238  DIFFERENTIAL   CALCULUS.  fART.  224. 

^^1+^2+^3=^,  (1) 

p  =  a-\-'b  -\-c=  2r  (sin  ^i  +  sin  O^.  +  sin  ^3) 

is  the  function  we  are  to  make  a  maximum,  and  is  a  function  of 
two  independent  variables,  say  0^  and  0^  ;  for  we  can  regard  ^^3  as 
depending  on  0i  and  6^  through  equation  (1).  As  r  is  a  con- 
stant, it  will  be  enough  to  make 

u  =  sin  <?!  4-  sin  0^  +  sin  ^^3  a  maximum. 

2>tjj  u  =  cos  ^1  -}-  cos  <?3  Z>0j  6*3 ; 

for,  since  ^1  +  <92  +  ^3  =  7^? 

changing  0i  without  changing  02  will  change  0^. 

hence  De  u  =  cos  0i  —  cos  0^. 

Dq^U  =  COS  ^2  —  COS  ^3, 

for  1)0/3= -1, 

2)^2^  =  —  sin  Ox  —  gin  ^3, 

De^De^u=  —  sin6'3, 
D^l  ^  _  _  gjjj  ^2  —  sin  6^. 
Make  De^i^  =  0  and  Db^u  =  0. 

cos  01  —  cos  ^3  =  0 
cos  02  —  cos  <?3  =  0 
0^  =  02=  0g, 


Chap.  XVI.]    MAXIMA  AND  MINIMA   OF   FUNCTIONS.         239 

Substitute  these  values  in  De^u,  &c.,  and 

Delu=  -2sin^yi  =  ^, 

De^  Dq^  u=  —  sin  0^  =  B, 

Delu=  -2  sin  6/1=  C, 

AC-B^  =  4sin2^i  -  sin^^i  =  3  sin2^i>0, 

A=  —  2sin^^i<0,    and  u  is  a  maximum. 

Since  0^=6^=6^, 

a=b  =  c; 

and  the  required  triangle  is  equilateral. 

(3)    To  inscribe  in  a  circle  a  triangle  of  maximum  area. 

Ans.    The  triangle  is  equilateral. 

225.  Ver}'  often  it  is  unnecessary  to  examine  the  second  de- 
rivatives, as  the  nature  of  the  problem  enables  one  to  determine 
whether  the  value  of  the  variables  obtained  b}^  writing  the  first 
derivatives  equal  to  zero  corresponds  to  maximum  or  minimum 
values  of  the  function. 

Examples. 

(1)  Required  the  form  of  a  parallelopiped  of  given  volume 
and  minimum  surface.  Ans.    A  cube. 

(2)  Required  the  form  of  a  parallelopiped  of  given  surface 
and  maximum  volume.  Ans.    A  cube. 

(3)  An  open  cistern  in  the  form  of  a  parallelopiped  is  to  be 
built,  capable  of  containing  a  given  volume  of  water,  what  must 
be  its  form  that  the  expense  of  lining  its  interior  surface  may  be 


a  mmimum  r 


Ans.   Length  and  breadth  each  double  the  depth. 


240  DIFFERENTIAL   CALCULUS.  [Art.  226. 


CHAPTER   XVII. 

THEORY  OF  PLANE  CURVES. 

Concavity  and  Convexity. 

226-  The  words  concavity  and  convexity  are  used  in  mathe- 
matics in  their  ordinary  sense.  A  curve  is  concave  toward  the 
axis  of  X  when  it  bends  toward  the  axis  ;  convex,  when  it 
bends  from  the  axis :  that  is,  when  in  passing  along  the 
curve  its  inclination  to  the  axis  of  X  decreases,  the  curve 
is  concave ;  when  it  increases,  the  curve  is  convex,  sup- 
posing that  the  portion  of  the  curve  considered  lies  above  the 
axis ;  if  it  lies  below  the  axis,  the  rule  just  given  must  be  re- 
versed.    We   have  seen  that  the  tangent  of  this  inclination, 

dij 
which  we  have  called  r,  is  equal  to  — .     If  the  curve  is  concave 

dx 

and  above  the  axis,  r  decreases  as  we  increase  x,  tanr  or  -i 

dx 

d^y 
decreases,  and  -t-c<0,  by  Art.  37.     If  the  curve  is  convex, 

d^y 

227.    A  point  at  which  the  curve  is  changing  from  convexity 

to  concavity,  or  from  concavity  to  convexity,  is  called  a  j^oint 

d^v 
of  inflection.     At  such  a  point,  — |  is  changing  from  a  negative 

ax 

to  a  positive  value,  or  from  a  positive  to  a  negative  value,  and 
consequently  must  be  passing  through  the  value  zero.     To  sum 

up:  if  y=fx 

is  the  equation  of  a  plane  curve,  at  any  point  corresponding  to 


Chap.  XVII.]        THEOKY  OF  PLANE  CURVES.  241 

a  value  of  x  that  makes  ^-^<0,  the  curve  is  concave  towards 
the  axis  of  ic,  if  above  the  axis ;  convex,  if  below.  At  any 
point  corresponding  to  a  value  of  x  that  makes  ^yi^^O,  the 
curve  is  convex  towards  the  axis  of  x^  if  above  the  axis  ;  con- 
cave, if  below.     Any  point  corresponding  to  a  value  of  x  that 

makes  — ^  =  0 

dx^ 

is  in  general  a  point  of  inflection. 
We  have  seen  that  the  curvature, 

dx" 
k  = 


It  is  easily  seen  that  at  a  point  of  inflection  this  value  changes 
sign. 

228.  These  same  tests  for  concavit}',  convexity,  and  inflec- 
tion can  be  very  simply  obtained  by  the  aid  of  Taylor's  Theorem. 

Let  y  =fx 

be  the  equation  of  a  curve,  and  let  it  be  required  to  discover 
whether  the  curve  is  concave  or  convex  toward  the  axis  of  X  at 
the  point  corresponding  to  the  value 

x=a. 

Draw  a  tangent  at  the  point  in  question,  and  erect  ordinates  to 
the  curve  and  to  the  tangent  near  the  point  of  contact. 

It  is  evident  that  the  ordinate  of  a  point  in  the  curve  minus 
the  ordinate  of  the  corresponding  point  of  the  tangent,  must  be 
negative  on  both  sides  of  the  point  of  contact,  if  the  curve  is 
concave^  and  positive  on  both  sides  of  the  point  of  contact,  if  the 


242 


DIFFERENTIAL   CALCULUS. 


[Art.  228. 


curve  is  convex.     If  the  point  is  sl  point  of  inflection^  this  differ 
ence  will  have  opposite  signs  on  different  sides  of  the  point. 


CONCAVE.  CONVEX.  INFLECTION. 

The  equation  of  the  tangent  at  the  point  corresponding  to 

x=  a 

is  y-fa=f'a(x-a),  by  Art.  28,  [1]. 

Let  x  =  a-\-h 

in  the  equations  of  curve  and  tangent,  and  call  the  con-esponding 
values  of  y,  ?/i  and  ?/2 ;  then 

2/2  =/«  +  //'«• 


by  Taylor's  Theorem. 


yi-'J.  =  ~f"a  +  ^^f"'(a  +  Oh). 


If /"a  does  not  equal  zero,  h  may  be  taken  so  small  that  the 
sign  of  2/i  —  2/2  will  be  the  sign  of  —f"a. 

Iff  "a  is  positive  this  sign  is  positive  whether  h  is  positive  or 
negative,  and  the  curve  is  convex.  Iff'a  is  negative,  2/1  —  2/2  is 
negative  both  before  and  after  x  =  a,  and  the  curve  is  concave. 

If  f"a  =  0  and  f"'a  does  not  vanish,  the  sign  of  2/1  —  2/2  will 
change  as  the  sign  of  h  changes,  and  we  shall  have  a  point  of 
inflection. 


Chap.  XVII.]        THEORY   OF   PLANE   CURVES.  243 

229.    E'or  example,  let  us  see  whether  the  curve 
a^  +  /=25 
is  convex  or  concave  towards  the  axis  of  X  at  the  point  (3,4). 
2xdx-\-2ydy=:0.  (1) 

2dx^  +  2dy^  +  2yd^y  =  0.  (2) 

From(l)  dy=-—. 

y 

Substitute  in  (2) ,  2dx'  +  ?^^  -f  2ydj'y  =  0, 

{x'  +  y')dx'-\-fd:'y  =  0, 
25dx'  +  ifdhj=0. 

dx"  f  64 

at  the  point  (3,4);  and  the  curve  is  concave. 
Again,  let  us  see  whether  the  curve 

y  =  x{x  —  aY    has  points  of  inflection 

dy 

—  =  (^x-ay-\-4x(x-a)\ 

d^y 


dx" 


—  %{x  —  ay  +  \2x{x  —  a)'% 


Write 


-^  =  ^Q{x-aY-^24x{x-a), 

d*y 

^  =  96  (a: -a) +  240' 


244  DIFFERENTIAL   CALCULUS.  L-^RT.  229. 

and  we  get  8{x  — ay -\-12x{x  —  ay=0 

or  2(x-ay-{-3x(x-ay^0. 

One  root  is  a;  =  a  ; 

divide  by  (x  —  ay,  and 

2aj  -  2a  +  3a;  =  0. 

2a 
a;  =  — -  IS  tne  remaining  root, 

o 

If  a,=  ^, 

5 

— I  does  not  equal  zero,  and  we  get  a  point  of  inflection. 
If  a;  =  a. 


d^ 


=  0, 


— ^  does  not  equal  zero,  and  the  point  is  not  a  point  of  inflection. 
ax 

Examples. 

there   is   a  point  of  inflection   at  the  origin,   and   also  when 
a;=±aV(3). 

(2)  ir  |  =  ^(«-^), 

there  is  a  point  of  inflection  when  x  =  — . 

(3)    If  xl  =  logy, 

there  is  a  point  of  inflection  when  a;  =  8. 


Chap.  XVII.]       THEOEY  OP  PLANE  CURVES. 
(4)    If 

there  is  a  point  of  inflection  when  x  =  ael. 


245 


CL 


Singular  Points, 

230.  Singular  points  of  a  curve  are  points  possessing  some 
pecuharity  independent  of  the  position  of  the  axes.  Such  points 
are,  — 

1.  Points  of  inflection  (Art.  228); 

2.  Multiple  points ; 

3.  Cusps ; 

4.  Conjugate  points ; 

5.  Points  d'arret ; 

6.  Points  saillant. 

231.  (2)  A  multiple  point  is  one  through  which  two  or  more 
branches  of  the  curve  pass.     If  only  two  branches  pass  through 


DOUBLE  POINT.         OSCULATING  POINT. 


CONJUGATE  POINT. 


POINT  D'ARRET. 


POINT  SAILLANT. 


the  point,  it  is  a  double  point.     A  double  point  at  which  the 
branches  of  the  curve  are  tangent  is  an  osculating  point. 

(3)  An  osculating  point  where  both  branches  of  the  curve 
stop  is  a  cusp. 

(4)  An  isolated  point  of  a  curve  is  a  conjugate  point. 


246  DIFFERENTIAL   CALCULUS.  [Art.  232. 

(5)  A  point  at  which  a  single  branch  of  a  curve  suddenty  stops 
is  a  poiiit  d'arrH. 

(6)  A  double  point  at  which  the  two  branches  of  the  curve 
stop  without  being  tangent  to  each  other  is  a  point  saillant. 


Multiple  Points. 

232.    At  a  multiple  point,  the  curve  will  in  general  have  more 

than  one  tangent,  and  therefore  —  will  have  more  than  one  value. 

dx 

Let  <p  =  0 

be  the  equation  of  the  curve  in  rational  algebraic  form. 

dx       Dy(p  "^ 

by  Art.  202.    For  any  given  values  of  x  and  y,  D^(p  and  Dy(p  will 
have  each  a  definite  value,  as  they  are  rational  polynomials  ;  and 

-^  will  have  but  one  value,  unless  D^^p  and  D^ip  are  both  zero, 
dx 

in  which  case  -^  =  -,  and  is  indeterminate ; 

dx     0 

hence,  our  fundamental  condition  for  the  existence  of  a  multiple 

point  is  D^(p  =  0   and   Dy<p  =  Q. 

To  determine  -^  in  that  case,  we  differentiate  numerator  and 
dx 

^^        Dj<p  +  D^ny<p^ 

denominator,  -^  = (1) 

dx 

Clearing  of  fractions  gives  us 


Chap.  XVII. ]        THEORY  OF  PLANE  CURVES.  247 

a  quadratic  to  tietermine  -^.     Unless  (1)  is  still  indeterminate, 
dx 

that  is,  unless  DJ<f,  D^Dyff^  and  Dy<p  are  all  zero,  we  get  two 

values  of  — ,  and  the  point  is  a  double  point, 
dx 

If  _^  is  still  indeterminate,  we  differentiate  (1)  again,  and  get 
dx 

dii 
to  determine  — .     We  have  then  three  tangents  at  the  point, 
dx 

which  is  a  triple  point. 


233.    If  the  values  of  ^  obtained  from  Art.  232  (2)  are  equal, 
dx 

the  two  tangents  at  the  double  point  coincide,  and  the  point  is 

an  osculating  i^oint  or  a  cusp;  and  we  cannot  tell  which  except 

by  actually  tracing  the  curve  in  the  neighborhood  of  the  point. 

dxi 
If  the  two  values  of  —  are  imaginary,  no  tangent  can  be 

OjX 

drawn  at  the  point,  which  is  then  a  conjugate  point. 

A  point  d'arret  or  a  2)oint  saillant  can  be  discovered  only  by 
inspection  when  attempting  to  trace  the  curve  ;  they  occur  only 
in  transcendental  curves. 


ExAivrPLE. 
234.   To  investigate  the  existence  of  multiple  points  in  the 
curve  x'^  —  a^x^  -f  a^i/  =  0. 

D^<P  =  4ar'—  2a^x, 

Dy(p  =  ^a^y-, 
D^^<p==l2a?-2a\ 


248  DIFFERENTIAL   CALCULUS.  [Art.  234. 

D^<p  and  Dy(p  must  equal  zero. 

if  a;  =  0  or  if  a;  =  ±  — ^. 

V(2) 

"la^y  =  0  if  2/  =  0  ; 
hence  x  must  equal  zero,  and  ?/  equal  0, 

or  x  =  ±  — ^   and  ?/  =  0  ; 

but  (  ±  ,0)  is  not  a  point  of  the  curve  ;  therefore  we  need 

consider  onl}^  (0,0) .     In  this  case, 

AV=2a^ 


2a^f^j  -2a2  =  o, 


\dx) 


1, 


ax 

and  the  origin  is  a  double  point  of  the  curve,  the  two  branches 
making  with  the  axis  of  X  angles  of  45°  and  135°  respectively. 


Chap.  XVII.]        THEORY   OF  PLANE  CURVES.  249 

Example. 
235.   Consider  x^-y^=0. 

Dy<p=-2y, 

Make  Sa^^  =  0  and  -  2?/  =  0. 

We  get  x  =  0 


as  the  only,  point  we  need 
y  =  0\      consider  liere. 


In  this  case,  D^(p  =  0, 

AV=-2. 


ax 


The  values  of  -^  are  equal,  and  the  origin  is  an  osculating  point, 


dy 
dx 
both  branches  being  there  tangent  to  the  axis  of  X. 

Since  y^  =  x^, 

it  is  easily  seen  that  the  curve  lies  to  the  right,  and  not  to  the 
left,  of  the  origin,  which  is  therefore  a  cusp. 


250  DIFFERENTIAL  CALCULUS.  [Art.  236. 

Example. 
236.  bcc^-3:^-hay"  =  0. 

D^<P  =  2bx-Sx^, 

Dy<p=2ay, 
D,^<P  =  2h-^x, 

2bx  -3x^=0 

2a?/ =  0 

x=0] 

>  is  to  be  considered. 

At  this  point,  D^^  <p  =  26, 

2a(^y+2b  =  0, 
\dxj 

fdyV_  ^ 
\dxj  ~~    a 

If  b  and  a  have  the  same  sign,  -1  is  imaginary,  and  the  origin 

dx 
is  a  conjugate  point ;  a  result  that  can  be  easil}^  verified  by  ex- 
amining the  equation. 

Examples. 

(1)    Show  that  the  curve  y  =  a;  log  a:  has  a  point  d'arret  at  the 
origin. 


Chap.  XVII. J        THEORY  OF  PLANE  CURVES.  251 

(2)  Show  that  the  curve  y  = ;  has  a  point  saillant  at  the 

origin,  and  find  the  directions  of  the  tangents  at  that  point. 

(3)  Show  that  {y  —  xY  =  ^  and  {y  —  ic^)-  =  ^  have  cusps  at 
the  origin. 

(4)  Show  that  (xy -\-\y-{-{x  —  \y  {x—2)  =  ()  has  a  cusp  at 
the  point  x  =  \. 

(5)  Show  that  x'^  —  aa^ y  —  axy^ -\- a^ y^  =  0   has  a  conjugate 
point  at  the  origin. 

(6)  Find  the  singular  points  in  the  following  curves  :  — 

{y  +  x-\-iy  =  {l-xy; 

y^  —  axy^  -f  a.''*  =  0  ; 

2/2  =  x^  —  X* ; 

y^  -j-  xy^  +  o(^(ax  —  by)  =  0. 


Contact  of  Curves. 

237.   Let  y=fx  and  y  =  Fx 

be  the  equations  of  two  curves.     If 

fa  =  Fa, 

the  curves  intersect  at  the  point  whose  abscissa  is  a.     If,  in 

addition,  F'a=f'a, 

the  tangents  at  this  point  of  intersection  coincide,  and  the  curves 
are  said  to  have  contact  at  the  point  in  question.     If 

fa  =  Fa,   F'a=f'a,    and   F"a  =  f'a, 


252  DIFFERENTIAL   CALCULUS.  [Art.  238. 

the  curves  have  contact  of  the  second  order  at  the  point.     If 

Fa=fa,   F'a=f'a,   F"a=/"a,  etc., -F^'^^a  =/<">«, 

the  curves  are  said  to  have  contact  of  the  nth  order  at  the  point 
whose  abscissa  is  a. 

Contact  of  a  higher  order  than  the  first  is  called  osculation. 

238.  The  difference  between  the  ordinates  of  points  of  the 
two  curves  having  the  same  abscissa  and  infinitely  near  the 
point  of  contact,  is  an  infinitesimal  of  an  order  one  higher  than 
the  order  of  contact  of  the  curves. 

Let  x=  a  -\-  Jx, 

2/i=/(«+^a;), 

and  2/2  =  ^(«  +  ^^)  ? 


y,=fa-\-Axfa  +  ifff"a  + 4.(M!/(«)a 

+  i^^f^^^'\a-^OJx), 
(w-hl)! 

2/2  =  Fa+JxF'a  -f  iM.'  F"a  + +  iM!  i^^a 

^(n  +  1)!     ^      ^ 
If  the  curves  have  contact  of  the  nth  order, 
Fa=fa, 
i?^'a  = /'a,  etc.,  i^("^ a  =/("^ a. 

(n  +  1)! 


% 


Chap.  XVtI.]        THEORY  OF  PLANE  CURVES.  253 

which  is  infinitesimal  of  the  (w  + 1 )  st  order,  if  Jx  is  an  infini- 
tesimal. It  follows,  then,  that  the  order  of  contact  indicates 
the  closeness  of  the  contact ;  that  is,  the  higher  the  order  of 
contact  of  two  curves,  the  less  rapidly  they  recede  from  each 
other  as  they  depart  from  the  point  of  contact. 

239 .  Let  it  be  requii  ed  to  find  the  equation  of  the  circle  having 
contact  of  the  second  order  with  the  curve 

y  =fx  at  the  point  (xi.yi) . 

Let  a  and  b  be  the  coordinates  of  the  centre,  and  r  the  radius 
of  the  required  circle.  Call  (X,Y)  any  point  of  the  required 
circle,  then  its  equation  is 

(X-ay-{-{Y-by  =  'f\ 
By  our  conditions. 


\dX'^)x=x^  \dxy^  =  ari ' 


but 


dY^      X-a 
dX  Y-b' 

fdY\  _      Xi  —  a 
\dXjr=:^^  Vi-b 

d^Y  -r" 


dX-"     (  Y-  by' 

fd'Y\      ^      -')^ 

\dxVx=~:,{y,-by' 


hence  fd^^  _x^-a   ' 

\dxjx=x.  Vi  —  b 


\d^jx=x^  {yi-by 


254  DIFFERENTIAL   CALCULUS. 

From  these  equations,  and 

we  can  get  the  required  values  of  a,  6,  and  r. 
Dropping  accents,  for  the  sake  of  simplicity, 


dPy      dx 


TArt.  239. 


substituting  in 


^x-ay  +  (y-by  =  7^ 


rtf^^Y=l4-^^A, 


dx'J  ■ 


dx 


1 


[ 


1  + 
1  + 


r=± 


dy 
dx 

df^ 
dx 


d^ 
dx" 


which  is  the  familiar  value  of  the  radius  of  curvature  of 

at  the  point  {x,y).  Hence,  our  osculating  circle  is  that  circle 
having  contact  of  the  second  order  with  the  given  curve  at  the 
point  in  question. 


Chap.  XVII.]        THEORY   OF   PLANE   CURVES.  255 

Examples. 

(1)  In  the  curve      2/  =  a?''  —  4a;^  —  18a^, 

show  that  the  radius  of  curvature  at  the  origin  is  3*^. 

(2)  Find  the  parabola  whose  axis  is  parallel  to  the  axis  of 

y,  which  has  the  closest  possible  contact  with  the  curve  y  =  — 

or 

at  the  point  where  x  =  a.  -d      .,     (       a\^     a  f       a\ 

Hesult.    [x ]  =-  [y . 

(3)  Prove  that,  if  the  order  of  contact  of  two  curves  is  even, 
they  cross  each  other  at  the  point  of  contact ;  if  odd,  the}'  do 
not  cross. 

Envelops. 

240.  If  the  equation  of  a  curve  contain  an  undetermined 
constant,  to  different  values  of  this  constant  will  correspond 
different  curves  of  a  series.  Such  an  equation  is  said  to  contain 
a  variable  parameter^  the  name  being  applied  to  a  quantity  which 
is  constant  for  any  one  curve  of  a  series,  but  varies  in  changing 
from  one  curve  to  another.     For  example  :  in  the  equation 

{x-ay  +  if  =  i^ 

let  a  be  a  variable  parameter ;  then  the  equation  represents  a 
series  of  circles,  all  having  the  radius  r,  and  all  having  their 
centres  on  the  axis  of  X. 

A  curve  tangent  to  each  of  such  a  series  of  curves  is  called  an 
envelop. 

241.  Two  curves  of  such  a  series  corresponding  to  two  differ- 
ent values  of  the  parameter  will  in  general  intersect.  If  they  are 
made  to  approach  each  other  indefinitely,  by  bringing  the  two 
values  of  the  parameter  nearer  together,  their  point  of  intersec- 
tion will  evidentl}'  approach  the  enveloping  curve,  which  then 
may  be  regarded  as  the  locus  of  the  limiting  position  of  a  point 


256  DIFFERENTIAL   CALCULUS.  [Art.  241. 

of  intersection  of  any  two  curves  of  the  series  as  the  curves  are 
made  to  indefiyiitely  approach.  From  this  point  of  view  the 
equation  of  an  envelop  is  easily  obtained. 

Let  f{x,y,a)^0  (1) 

be  the  given  equation  of  the  series  of  curves,  a  being  a  variable 

parameter.  f{x,y,a  -f  Ja)  =  0  (2) 

will  be  any  second  curve  of  the  series.     The  equation 

f(x,y,a  +  Ja)  -f{x,y,a)  =  0  (3) 

represents  some  curve  passing  through  all  the  points  of  intersec- 
tion of  (1)  and  (2)  by  the  principle  in  Anal^^tic  Geometry :  "If 
u=0  and  v  =  0  are  the  equations  of  two  curves,  u-\-kv  =  0  rep- 
resents a  curve  containing  all  their  points  of  intersection,  and 
having  no  other  point  in  common  with  them." 

f(x,y,a  -f  Ja)  -f{x,y,a)  _ 
da  ^ 

is  equivalent  to  (3) .     If,  now,  Ja  be  decreased  indefinitely, 


limit 


or 


'f{x,y,a+Aa)-f{x,y,a)~\  _ 

Aa  J       ^' 

Daf{x,y,a)=0,  (4) 


contains  the  limiting  position  of  the  point  of  intersection  of  (1) 
and  (2).  Let  (x'^y')  be  this  point,  and  therefore  any  point  of 
the  required  locus.     Since  {x',y')  is  on  (4),  and  also  on  (1), 

Daf(x',  y\  a)  =  0    and  f{x',  y\  «)  =  0  ; 

we  can  eliminate  a  between  these  equations,  and  we  shall  hav< 
a  single  equation  between  x'  and  y\  which  will  be  the  equation 
of  the  required  envelop. 


Chap.  XVII.]       THEORY  OF  PLANE  CURVES.  267 

242.  For  example  :  let  us  find  the  envelop  of 

{x-oy^f-i^  =  o,  (1) 

a  being  a  variable  parameter. 

X-o.=zQ.  (2) 

Eliminating  «  between  (1)  and  (2),  we  get 

the  equation  of  a  pair  of  straight  lines  parallel  to  the  axis  of  X, 
as  the  required  envelop. 

243.  When  dealing  with  the  properties  of  evolutes,  we  proved 
that  ever}'  normal  to  the  original  curve  must  be  tangent  to  the 
evolute.  We  ought,  then,  to  be  able  to  find  the  e volute  of  any 
curve  by  treating  it  as  the  envelop  of  the  normals  of  the  curve.    f|| 

Let  y=fa 

be  the  equation  of  the  original  curve 

is  the  equation  of  the  normal,  or 

(^yy-y,)  +  x-x,  =  0.  (1) 

Xq  is  the  variable  parameter, 

^'o/=g(^-^o)-(gJ-l  =  0,  (2) 


258  DIFFERENTIAL  CALCULUS.  [Abt.  243. 


dxQ 


\dxQj 


dxQ 
but  yo=fxo, 

and  we  must  eliminate  Xq  and  2/01  by  the  aid  of  these  three  equa- 
tions, to  obtain  the  equation  of  the  evolute.  These  equations 
are  the  ones  obtained  by  a  different  method  in  Art.  93. 

Examples. 

(1)  Find  the  envelop  of  all  elhpses  having  constant  area,  the 
axes  being  coincident. 

Result.    A  pair  of  equilateral  h3^perbolas. 

(2)  A  straight  line  of  given  length  moves  with  its  extremities 
on  the  two  axes,  required  its  envelop.      Result.   x^-\-y^  =  a^. 

(3)  Find  the  envelop  of  straight  lines  drawn  perpendicular  to 
the  normals  to  a  parabola  y^  =  4aa;  at  the  points  where  they  cut 
the  axis.  Result,   y^  =  4:a{2a  —  x) . 

(4)  Circles  are  described  on  the  double  ordinates  of  a  parab- 
ola as  diameters.     Show  that  their  envelop  is  an  equal  parabola. 

(5)  Find  the  envelop  of  all  ellipses  having  the  same  centre, 
and  in  which  the  straight  line  joining  the  ends  of  the  axes  is  of 
constant  length.  Result,    x  ±y  =  ±c. 

(6)  Show  that  the  envelop  of  a  circle  on  the  focal  radius  of  an 
ellipse  as  diameter  is  the  circle  on  the  major  axis. 


I 


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A  TREATISE   ON   PLANE  SURVEYING. 

By  DANIEL  CARHART,  C.E.,  Professor  of  Civil  Engineering 
in  the  Western  University  of  Pennsylvania,  Alleghany. 

Octavo.     Half  leather.     536  pages.  Introduction  Price,  $i.8a 

This  work  covers  the  ground  of  Plane  Surveying.  It  illus- 
trates and  describes  the  instruments  employed,  their  adjust- 
ments and  uses ;  it  exemplifies  the  best  methods  of  solving  the 
ordinary  problems  occurring  in  practice,  and  furnishes  solutions 
for  many  special  cases  which  not  infrequently  present  them- 
selves. It  is  the  result  of  twenty  years'  experience  in  the  field 
and  technical  schools,  and  the  aim  has  been  to  make  it  ex- 
tremely practical,  having  in  mind  always  that  to  become  a 
reliable  surveyor  the  student  needs  frequently  to  manipulate 
the  various  surveying  instruments  in  the  field,  to  solve  many 
examples  in  the  class-room,  and  to  exercise  good  judgment  in 
all  these  operations.  Not  only,  therefore,  are  the  different 
methods  of  surveying  treated,  and  directions  for  using  the 
instruments  given,  but  these  are  supplemented  by  various  field 
exercises  to  be  performed,  by  numerous  examples  to  be  wrought, 
and  by  many  queries  to  be  answered. 

Chapter       I.  Chain  Surveying. 

"  11.  Compass  and  Transit  Surveying. 

«  III.  Declination  of  the  Needle. 

**  IV.  Laying  Out  and  Dividing  Land. 

"  V.  Plane  Table  Surveying. 

"  VI.  Government  Surveying. 

"  VII.  City  Surveying.     Including  the  Principles  of  Levelling. 

**  VIII.  Mine  Surveying.   Including  the  Elements  of  Topography. 

A  Table  of  Logarithms  of  Numbers;  a  Table  of  Natural 
and  Logarithmic  Sines,  Cosines,  Tangents,  and  Cotangents; 
a  Traverse  Table  ;  and  many  others. 


GrNN  &  COMPANY,  Publishers,  Boston,  New  York,  and  Chicago. 


PLANE   AND   SOLID 

Analytic  Geometry 

By  Frederick  H,  Bailey,  A.M.  (Harvard),  and  Frederick 

S.  Woods,  Ph.D.  (Gottingen),  Assistant  Professors 

of  Mathematics  in  Massachusetts  Institute 

of  Technology. 


8vo.  Cloth.  371  pages.  For  introduction,  $2.00. 


'TTHIS  book  is  intended  for  students  beginning  the  study 
of  analytic  geometry,  primarily  for  students  in  colleges 
and  technical  schools.  While  the  subject-matter  has  been 
confined  to  that  properly  belonging  to  a  first  course,  the 
treatment  of  all  subjects  discussed  has  been  complete  and 
rigorous.  More  space  than  is  usual  in  text-books  has  been 
devoted  to  the  more  general  forms  of  the  equations  of  the 
first  and  the  second  degrees.  The  equations  of  the  conic 
sections  have  been  derived  from  a  single  definition,  and  after 
the  simplest  types  of  these  equations  have  been  deduced,  the 
student  is  taught  by  the  method  of  translation  of  the  origin 
to  handle  any  equation  of  the  second  degree  in  which  the 
X  y  term  does  not  appear.  In  particular,  the  equations  of 
the  tangent,  the  normal,  and  the  polar  have  been  determined 
for  such  an  equation.  Only  later  is  the  general  equation  of 
the  second  degree  fully  discussed. 

In  the  solid  geometry,  besides  the  plane  and  the  straight 
line,  the  cylinders  and  the  surfaces  of  revolution  have  been 
noticed,  and  all  the  quadric  surfaces  have  been  studied  from 
their  simplest  equations.  This  study  includes  the  treatment 
of  tangent,  polar,  and  diametral  planes,  conjugate  diameters, 
circular  sections,  and  rectilinear  generators. 

Throughout  the  work  no  use  is  made  of  determinants  or 

calculus. 

•         • — • 

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ELEMENTS  OF  PLANT  ANATOMY. 

By  Emily  L.  Gregory,  Professor  of  Botany  in  Barnard  College.  148  pages. 
Illustrated.     For  introduction,  $1.25. 

Designed  as  a  text-book  for  students  who  have  already  some  knowledge  of  general 
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ELEMENTS  OF  STRUCTURAL  AND  SYSTEMATIC  BOTANY. 

For  High  Schools  and  Elementary  College  Courses.  By  Douglas  H.  Campbell, 
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PLANT  ORGANIZATION. 

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The  size  and  general  features  of  the  Introduction  to  Physical 
Science  in  its  present  revised  form  have  been  changed  little, 
but  numerous  slight  changes  have  been  made  throughout 
the  work  which  will  be  found  improvements  and  which  will 
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The  leading  feature  of  the  Elements  of  Physics  is  that  it  is 
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tions, and  precede  the  statements  of  principles  and  laws. 

Physical  Experiments  contains  the  laboratory  exercises 
required  for  admission  to  Harvard  University  and  to  many 
other  colleges.  Specific  directions  are  given  for  the  prepa- 
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Exercises  and  problems  follow  the  discussion  of  laws  and 
principles. 

The  subject-matter  is  so  divided  that  the  book  can  be 
used  by  advanced  schools,  or  by  elementary  ones  in  which 
the  time  allotted  to  chemistry  is  short. 

Chemical  Experiments  is  for  the  use  of  students  in  the 
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The  Laboratory  Manual  contains  one  hundred  sets  of  ex- 
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ASTRONOriY 


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A  Series  of  text-books  on  astronomy  for  high  schools,  academies,  and  colleges. 

Prepared  by  one  of  the  most  distinguished  astronomers  of  the  world, 

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Lessons  in  Astronomy.  Including  Uranography.  Revised  Edition. 
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duction, $2.25. 

The  Lessons  in  Astronomy  (recently  brought  up  to  date)  was 
prepared  for  schools  that  desire  a  brief  course  free  from 
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pains  has  been  taken  not  to  sacrifice  accuracy  and  truth 
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to  the  present  time. 

The  Elements  of  Astronomy  is  an  independent  work,  and 
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been  paid  to  making  all  statements  correct  and  accurate  so 
far  as  they  go. 

The  eminence  of  Professor  Young  as  an  original  investi- 
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and  an  instructor  in  college  classes,  led  the  publishers  to 
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It  is  conceded  to  be  the  best  astronomical  text-book  of  its 
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Inductivk   Logic 


By    WM.    G.    BALLANTINE, 

President  of  Oberlin  College. 


12mo.  Cloth.  viii  + 174  pages.  For  introduction,  80  cents. 

Although  this  is  preeminently  a  scientific  age,  surprisingly 
little  attention  is  given  in  our  colleges  and  universities  to  the  study 
of  inductive  logic.  The  neglect  is  probably  due  to  the  lack  of  a 
satisfactory  text-book.  Such  manuals  as  are  found  are  hardly 
more  than  meager  abridgments  of  some  chapters  of  the  great 
but  unequal  work  of  Mill. 

It  is  the  aim  of  this  book  to  present,  within  reasonable 
compass,  a  fresh  and  independent  statement  of  the  fundamental 
principles  of  inductive  logic,  consistently  carried  out  in  detail  and 
amply  illustrated  by  extracts  from  a  wide  range  of  philosophical 
and  scientific  writers.  The  best  modern  teachers  make  large  use 
of  the  library  and,  while  setting  forth  their  own  views,  seek  to 
acquaint  their  students  with  the  literature  of  the  subject  and  the 
history  of  opinion.  It  is  believed  that  these  numerous  quotations 
from  Bacon,  Mill,  Darwin,  Helmholtz,  G.  F.  Wright,  and  others, 
while  exactly  in  point  as  illustrations  and  elucidations,  will  also 
be  found  strikingly  interesting  in  themselves  and  highly  useful  in 
familiarizing  the  reader  with  the  phraseology,  literary  styles  and 
modes  of  thinking  of  those  eminent  authorities. 

Teachers  of  inductive  logic  will  be  pleased  to  find  here  a 
simple  account  of  the  relations  of  induction  and  deduction  which 
discards  the  notion  of  two  separate  realms  of  thought  in  one  of 
which  it  is,  and  in  the  other  is  not,  legitimate  to  draw  a  conclusion 
wider  than  the  premises.  The  classification  of  inductions  under 
three  heads,  as  primary,  secondary  and  mixed,  clears  away  the 
confusions  which  liave  arisen  from  the  attempt  to  bring  all  induc- 
tions under  a  single  definition.  The  doctrine  of  Causation  is  treated 
with  great  thoroughness,  but  the  notion  of  cause  is  not  made,  as 
in  Mill's  system,  the  root  of  the  whole  theory  of  induction. 


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Glaciers  of  North  America 

By  ISRAEL  C.  RUSSELL, 

Professor  of  Geology  in  the  University  of  Alichigan, 
A  uthor  of  "  Lakes  of  North  A  tnerica." 


8vo.  Cloth.  X  +  210  pages.  Illustrated. 
By  mail,  postpaid,  $1.90 ;  to  teachers  and  for  introdnotion,  $1.75. 


Recent  explorations  have  shown  that  North  America 
contains  thousands  of  glaciers,  some  of  which  are  not 
only  vastly  larger  than  any  in  Europe,  but  belong  to 
types  of  ice  bodies  not  there  represented.  In  the  study 
of  the  glaciers  of  North  America,  and  especially  of  those 
in  Alaska,  Professor  Russell  has  taken  an  active  part, 
and  this  book  not  only  presents  the  results  of  his  own 
explorations,  but  a  condensed  and  accurate  statement 
of  the  present  status  of  glacial  investigations.  Its 
popular  character  and  numerous  illustrations  will  make 
it  of  interest  to  the  general  reader. 


Edwin  S.  Balch,  Vice-President  Geographical  Society  of  Philadelphia : 
It  is  by  all  odds  the  clearest  work  on  glaciers  in  general  that  I  have 
ever  seen. 

F.  Bascom,  Department  of  Geology,  Bryn  Mawr  College,  Pa. :  No  one 
is  better  fitted  than  Professor  Russell,  by  extended  personal  explo- 
rations, to  discuss  with  authority  this  subject.  His  volume  combines 
in  an  unusual  degree  a  clearness  and  fascination  of  style  with  a  thor- 
oughly scientific  treatment. 

Henry  F.  Osborne,  American  Museum  of  N^atural  History,  New  York, 
N.  V. :  1  consider  it  a  contribution  of  very  great  value.  By  bringing 
together  all  that  is  at  present  known  upon  the  glaciers  of  North 
America  in  convenient  form,  this  work  will  stimulate  the  study  of  our 
own  living  glaciers  among  American  geological  students  and  thus 
accomplish  a  marked  service  to  American  geology. 


Department  of  Special  Publication. 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

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